At the risk of perhaps oversimplifying the problem, is there any reason that you wish to avoid using the proprietary Nvidia drivers?
The 340xx are no longer available via the official Manjaro repositories, however they are still obtainable in other formats:
- Download Display Driver 340.1080 drivers directly from Nvidia (not officially supported, but still a workable solution).
- Or possibly 340xx LTS via the AUR.
- Also [HowTo] get legacy 340xx NVIDIA drivers back seems promising.
Any of these options would doubtless come with caveats, being that the GPU (and machine, generally) is so dated.
Informational:-
The Sony Vaio VPCF115FM manual; in case it’s useful:
Downloadable version (.pdf):