Where to install the bootloader? (Dual boot with Windows 10)

Hi,
I have a laptop with dual boot Windows 10 and Ubuntu, when I broke my Ubuntu install I wanted to change to Manjaro, but I don’t remember what option(partition) I selected to install GRUB when installing Ubuntu, now I want to install Manjaro without losing any data from my Windows 10 partition, I just want to format the Ubuntu partition and install Manjaro there, without touching in Windows installation.
My question is where I will install boot loader? “/dev/sdb”, “/” or anything else? Could you guys please help me? My laptop use BIOS and the partition table is MBR. Above more information of what I have today:

============================== Boot Info Summary ===============================

 => Syslinux MBR (5.00 and higher) is installed in the MBR of /dev/sda.
 => Grub2 (v2.00) is installed in the MBR of /dev/sdb and looks at sector 1 of 
    the same hard drive for core.img. core.img is at this location and looks 
    for (,msdos5)/boot/grub. It also embeds following components:
    
    modules
    ---------------------------------------------------------------------------
    fshelp ext2 part_msdos biosdisk
    ---------------------------------------------------------------------------

sda1: __________________________________________________________________________

    File system:       vfat
    Boot sector type:  SYSLINUX 6.04
    Boot sector info:  Syslinux looks at sector 32784 of /dev/sda1 for its 
                       second stage. The integrity check of Syslinux failed. 
                       No errors found in the Boot Parameter Block.
    Operating System:  
    Boot files:        /boot/grub/grub.cfg /syslinux.cfg 
                       /efi/BOOT/grubx64.efi /efi/BOOT/mmx64.efi /ldlinux.sys

sdb1: __________________________________________________________________________

    File system:       ntfs
    Boot sector type:  Windows Vista: NTFS
    Boot sector info:  No errors found in the Boot Parameter Block.
    Operating System:  
    Boot files:        /BOOTMGR /BOOT/BCD

sdb2: __________________________________________________________________________

    File system:       ntfs
    Boot sector type:  Windows 8/2012: NTFS
    Boot sector info:  No errors found in the Boot Parameter Block.
    Operating System:  
    Boot files:        /bootmgr /Boot/BCD

sdb3: __________________________________________________________________________

    File system:       ntfs
    Boot sector type:  Windows 7/2008: NTFS
    Boot sector info:  No errors found in the Boot Parameter Block.
    Operating System:  Windows 10
    Boot files:        /Windows/System32/winload.exe

sdb4: __________________________________________________________________________

    File system:       Extended Partition
    Boot sector type:  Unknown
    Boot sector info: 

sdb5: __________________________________________________________________________

    File system:       ext4
    Boot sector type:  -
    Boot sector info: 
    Operating System:  Ubuntu 20.10
    Boot files:        /boot/grub/grub.cfg /etc/fstab /etc/default/grub 
                       /boot/grub/i386-pc/core.img

sdb6: __________________________________________________________________________

    File system:       swap
    Boot sector type:  -
    Boot sector info: 


================================ 4 OS detected =================================

OS#1:   Windows Recovery Environment (boot) on sdb1
OS#2:   Windows 10 (boot) on sdb2
OS#3:   Ubuntu 20.10 on sdb5
OS#4:   Windows 10 on sdb3

============================ Architecture/Host Info ============================

CPU architecture: 64-bit
Live-session OS is Ubuntu 64-bit (Ubuntu 20.04.2 LTS, focal, x86_64)


===================================== UEFI =====================================

This live-session is not in EFI-mode.
This session has been detected as 'live' because df -Th / contains overlay



============================= Drive/Partition Info =============================

Disks info: ____________________________________________________________________

sdb	: notGPT,	no-BIOSboot,	has-noESP, 	not-usb,	not-mmc, has-os,	2048 sectors * 512 bytes

Partitions info (1/3): _________________________________________________________

sdb1	: is-os,	32, nopakmgr,	no-docgrub,	nogrub,	nogrubinstall,	no-grubenv,	noupdategrub,	not-far
sdb2	: is-os,	32, nopakmgr,	no-docgrub,	nogrub,	nogrubinstall,	no-grubenv,	noupdategrub,	not-far
sdb3	: is-os,	32, nopakmgr,	no-docgrub,	nogrub,	nogrubinstall,	no-grubenv,	noupdategrub,	farbios
sdb5	: is-os,	64, apt-get,	grub-pc ,	grub2,	grub-install,	grubenv-ok,	update-grub,	farbios

Partitions info (2/3): _________________________________________________________

sdb1	: isnotESP,	part-has-no-fstab,	no-nt,	no-winload,	recovery-or-hidden,	BOOTMGR,	is-winboot
sdb2	: isnotESP,	part-has-no-fstab,	no-nt,	no-winload,	no-recov-nor-hid,	bootmgr,	is-winboot
sdb3	: isnotESP,	part-has-no-fstab,	no-nt,	haswinload,	no-recov-nor-hid,	no-bmgr,	notwinboot
sdb5	: isnotESP,	fstab-without-efi,	no-nt,	no-winload,	no-recov-nor-hid,	no-bmgr,	notwinboot

Partitions info (3/3): _________________________________________________________

sdb1	: not-sepboot,	no-boot,	part-has-no-fstab,	not-sep-usr,	no---usr,	part-has-no-fstab,	std-grub.d,	sdb
sdb2	: not-sepboot,	no-boot,	part-has-no-fstab,	not-sep-usr,	no---usr,	part-has-no-fstab,	std-grub.d,	sdb
sdb3	: not-sepboot,	no-boot,	part-has-no-fstab,	not-sep-usr,	no---usr,	part-has-no-fstab,	std-grub.d,	sdb
sdb5	: not-sepboot,	with-boot,	fstab-without-boot,	not-sep-usr,	with--usr,	fstab-without-usr,	std-grub.d,	sdb

fdisk -l (filtered): ___________________________________________________________

Disk sda: 14.42 GiB, 15472047104 bytes, 30218842 sectors
Disk identifier: 0x038b643c
      Boot Start      End  Sectors  Size Id Type
sda1  *     2048 30218841 30216794 14.4G  c W95 FAT32 (LBA)
Disk sdb: 298.9 GiB, 320072933376 bytes, 625142448 sectors
Disk identifier: 0x
      Boot     Start       End   Sectors   Size Id Type
sdb1            2048  25167871  25165824    12G 27 Hidden NTFS WinRE
sdb2  *     25169920  25374719    204800   100M  7 HPFS/NTFS/exFAT
sdb3        25374720 517724159 492349440 234.8G  7 HPFS/NTFS/exFAT
sdb4       517724160 625141759 107417600  51.2G  5 Extended
sdb5       517726208 621045759 103319552  49.3G 83 Linux
sdb6       621047808 625141759   4093952     2G 82 Linux swap / Solaris

parted -lm (filtered): _________________________________________________________

sda:15.5GB:scsi:512:512:msdos:Kingston DataTraveler 3.0:;
1:1049kB:15.5GB:15.5GB:fat32::boot, lba;
sdb:320GB:scsi:512:512:msdos:ATA Hitachi HTS54503:;
1:1049kB:12.9GB:12.9GB:ntfs::msftres;
2:12.9GB:13.0GB:105MB:ntfs::boot;
3:13.0GB:265GB:252GB:ntfs::;
4:265GB:320GB:55.0GB:::;
5:265GB:318GB:52.9GB:ext4::;
6:318GB:320GB:2096MB:linux-swap(v1)::;

blkid (filtered): ______________________________________________________________

NAME   FSTYPE   UUID                                 PARTUUID                             LABEL           PARTLABEL
sda                                                                                                       
└─sda1 vfat     64DB-E079                            038b643c-01                          UBUNTU 20_0     
sdb                                                                                                       
├─sdb1 ntfs     6C9EA9B79EA979EA                     300560a6-01                          PQSERVICE       
├─sdb2 ntfs     D6327BE7327BCACF                     300560a6-02                          SYSTEM RESERVED 
├─sdb3 ntfs     3C027DAB027D6B32                     300560a6-03                          Acer            
├─sdb4                                               300560a6-04                                          
├─sdb5 ext4     86b48f87-036d-4de4-8f2e-2e7db0ea9821 300560a6-05                                          
└─sdb6 swap     f477108e-97d1-47cd-95ae-81ed319405a2 300560a6-06                                          

df (filtered): _________________________________________________________________

                   Avail Use% Mounted on
sda1               11.7G  19% /cdrom
sdb1              684.1M  94% /mnt/boot-sav/sdb1
sdb2               64.3M  36% /mnt/boot-sav/sdb2
sdb3               22.5G  90% /mnt/boot-sav/sdb3
sdb5                5.2G  84% /mnt/boot-sav/sdb5

Mount options: __________________________________________________________________

sda1              ro,noatime,fmask=0022,dmask=0022,codepage=437,iocharset=iso8859-1,shortname=mixed,errors=remount-ro
sdb1              rw,relatime,user_id=0,group_id=0,allow_other,blksize=4096
sdb2              rw,relatime,user_id=0,group_id=0,allow_other,blksize=4096
sdb3              rw,relatime,user_id=0,group_id=0,allow_other,blksize=4096
sdb5              rw,relatime

====================== sda1/boot/grub/grub.cfg (filtered) ======================

Ubuntu
Ubuntu (safe graphics)
OEM install (for manufacturers)
Boot from next volume
UEFI Firmware Settings

========================= sda1/syslinux.cfg (filtered) =========================

DEFAULT loadconfig

LABEL loadconfig
  CONFIG /isolinux/isolinux.cfg
  APPEND /isolinux/

==================== sda1: Location of files loaded by Grub ====================

           GiB - GB             File                                 Fragment(s)
            ?? = ??             boot/grub/grub.cfg                             1

================== sda1: Location of files loaded by Syslinux ==================

           GiB - GB             File                                 Fragment(s)
            ?? = ??             syslinux.cfg                                   1
            ?? = ??             ldlinux.sys                                    1

====================== sdb5/boot/grub/grub.cfg (filtered) ======================

Ubuntu   86b48f87-036d-4de4-8f2e-2e7db0ea9821
Ubuntu, com o Linux 5.8.0-50-generic   86b48f87-036d-4de4-8f2e-2e7db0ea9821
Ubuntu, com o Linux 5.8.0-48-generic   86b48f87-036d-4de4-8f2e-2e7db0ea9821
Windows Recovery Environment (em sda1)   6C9EA9B79EA979EA
Windows 10 (em sda2)   D6327BE7327BCACF
### END /etc/grub.d/30_os-prober ###
### END /etc/grub.d/30_uefi-firmware ###

========================== sdb5/etc/fstab (filtered) ===========================

# <file system> <mount point>   <type>  <options>       <dump>  <pass>
# / was on /dev/sdb5 during installation
UUID=86b48f87-036d-4de4-8f2e-2e7db0ea9821 /               ext4    errors=remount-ro 0       1
# swap was on /dev/sdb6 during installation
UUID=f477108e-97d1-47cd-95ae-81ed319405a2 none            swap    sw              0       0

======================= sdb5/etc/default/grub (filtered) =======================

GRUB_DEFAULT=0
GRUB_TIMEOUT_STYLE=hidden
GRUB_TIMEOUT=10
GRUB_DISTRIBUTOR=`lsb_release -i -s 2> /dev/null || echo Debian`
GRUB_CMDLINE_LINUX_DEFAULT="quiet splash"
GRUB_CMDLINE_LINUX="locale=pt_BR"

Thanks.

Before doing anything else, I’d highly recommend you make yourself a backup and familiarise yourself with the process of restoring it, just in case. From within Windows 10, you can use a free edition of Macrium Reflect (not open source, however, it is the only Windows backup tool I have used and it seems to work pretty well). Make sure your backup is located on an external drive of some sorts to keep it safe. You should probably create Macrium’s bootable recovery media as well if you go this route. Test to make sure you can actually boot from it before proceeding.

As for boot loaders and such, I’m not experienced enough to say for certain what will happen. However, the Manjaro installer will give you the option to delete a partition (in your case, Ubuntu is sdb5) to make room for Manjaro. I assume Grub would also get installed to sdb. In my case it detected Windows 10 and added a working boot menu option for it. However, after installing Manjaro to the same drive as Windows 10, my laptop booted straight back to Windows 10. I think I had to change my UEFI boot order to get Grub to load.

can you boot on USB iso manjaro
and returns

inxi -Fza
System:    Kernel: 5.10.30-1-MANJARO x86_64 bits: 64 compiler: gcc v: 10.2.0 
           parameters: BOOT_IMAGE=/boot/vmlinuz-x86_64 lang=en_US keytable=us tz=UTC misobasedir=manjaro 
           misolabel=MANJARO_KDE_2102 quiet systemd.show_status=1 apparmor=1 security=apparmor driver=free nouveau.modeset=1 
           i915.modeset=1 radeon.modeset=1 
           Desktop: KDE Plasma 5.21.4 tk: Qt 5.15.2 wm: kwin_x11 vt: 1 dm: SDDM Distro: Manjaro Linux base: Arch Linux 
Machine:   Type: Laptop System: Acer product: Aspire 4745 v: V1.15 serial: <filter> 
           Mobo: Acer model: ZQ1 serial: <filter> BIOS: INSYDE v: 1.15 date: 06/04/2010 
Battery:   ID-1: BAT1 charge: N/A condition: N/A volts: N/A model: BAT1 type: Li-ion serial: <filter> status: Unknown 
CPU:       Info: Dual Core model: Intel Core i3 M 330 bits: 64 type: MT MCP arch: Nehalem family: 6 model-id: 25 (37) 
           stepping: 2 microcode: 11 cache: L2: 3 MiB 
           flags: lm nx pae sse sse2 sse3 sse4_1 sse4_2 ssse3 vmx bogomips: 17022 
           Speed: 1463 MHz min/max: 933/2133 MHz Core speeds (MHz): 1: 1463 2: 1463 3: 1463 4: 1463 
           Vulnerabilities: Type: itlb_multihit status: KVM: VMX disabled 
           Type: srbds status: Not affected 
           Type: tsx_async_abort status: Not affected 
Graphics:  Device-1: Intel Core Processor Integrated Graphics vendor: Acer Incorporated ALI driver: i915 v: kernel 
           bus-ID: 00:02.0 chip-ID: 8086:0046 class-ID: 0300 
           Device-2: ALi Gateway Webcam type: USB driver: uvcvideo bus-ID: 1-1.1:3 chip-ID: 0402:9665 class-ID: 0e02 
           Display: x11 server: X.Org 1.20.11 compositor: kwin_x11 driver: loaded: intel unloaded: modesetting 
           alternate: fbdev,vesa display-ID: :0 screens: 1 
           Screen-1: 0 s-res: 1920x1080 s-dpi: 96 s-size: 506x285mm (19.9x11.2") s-diag: 581mm (22.9") 
           Monitor-1: HDMI1 res: 1920x1080 hz: 60 dpi: 102 size: 480x270mm (18.9x10.6") diag: 551mm (21.7") 
           OpenGL: renderer: Mesa DRI Intel HD Graphics (ILK) v: 2.1 Mesa 21.0.2 direct render: Yes 
Audio:     Device-1: Intel 5 Series/3400 Series High Definition Audio vendor: Acer Incorporated ALI driver: snd_hda_intel 
           v: kernel bus-ID: 00:1b.0 chip-ID: 8086:3b56 class-ID: 0403 
           Sound Server-1: ALSA v: k5.10.30-1-MANJARO running: yes 
           Sound Server-2: JACK v: 0.125.0 running: no 
           Sound Server-3: PulseAudio v: 14.2 running: yes 
           Sound Server-4: PipeWire v: 0.3.25 running: no 
Network:   Device-1: Qualcomm Atheros AR8151 v1.0 Gigabit Ethernet vendor: Acer Incorporated ALI driver: atl1c v: kernel 
           port: 2000 bus-ID: 01:00.0 chip-ID: 1969:1073 class-ID: 0200 
           IF: enp1s0 state: down mac: <filter> 
           Device-2: Qualcomm Atheros AR928X Wireless Network Adapter vendor: Foxconn T77H047.31 802.11bgn driver: ath9k 
           v: kernel port: 2000 bus-ID: 02:00.0 chip-ID: 168c:002a class-ID: 0280 
           IF: wlp2s0 state: up mac: <filter> 
Drives:    Local Storage: total: 312.5 GiB used: 0 KiB (0.0%) 
           SMART Message: Unable to run smartctl. Root privileges required. 
           ID-1: /dev/sda maj-min: 8:0 vendor: Hitachi model: HTS545032B9A300 size: 298.09 GiB block-size: physical: 512 B 
           logical: 512 B speed: 3.0 Gb/s rotation: 5400 rpm serial: <filter> rev: C60F scheme: MBR 
           ID-2: /dev/sdb maj-min: 8:16 type: USB vendor: Kingston model: DataTraveler 3.0 size: 14.41 GiB block-size: 
           physical: 512 B logical: 512 B serial: <filter> scheme: MBR 
           SMART Message: Unknown USB bridge. Flash drive/Unsupported enclosure? 
Partition: Message: No partition data found. 
Swap:      Alert: No swap data was found. 
Sensors:   System Temperatures: cpu: 48.0 C mobo: N/A 
           Fan Speeds (RPM): N/A 
Info:      Processes: 183 Uptime: 5m wakeups: 4 Memory: 3.52 GiB used: 1.87 GiB (53.2%) Init: systemd v: 247 tool: systemctl 
           Compilers: gcc: N/A Packages: pacman: 1217 lib: 326 flatpak: 0 Shell: Bash v: 5.1.0 running-in: konsole 
           inxi: 3.3.04