Asus M5A78L-M-USB3 Amd Motherboard supported processor

Solution
Yes, but depending on how old the board is you may need to update the bios for FX processors.
And do not overclock, as the board does not have VRM heatsinks.
http://www.asus.com/us/supportonly/M5A78L-MUSB3/HelpDesk_CPU/
Support, in the sense that the chip will fit, boot, and run, yes.

Support in the sense of use for a half-decent gaming system, no. (for the FX8320)

That board will run a FX6300 at stock quite well, the problem is that to get the best out of a FX these days, you HAVE to overclock it. If you are using the system for home, personal, or business use, OK. If you want to game, render, edit, or any high-performance application NOT OK.

That board is totally unsuitable for any FX8??? chip.

As mentioned above, the motherboard lacks heatsinks on the power delivery system (VRM) and when the CPU gets going close to full load, the system with throttle down as the motherboard overheats.

If you plan on a FX8320 and are in the USA, for $75, you can get a Gigabyte GA 970a-UD3P which will handle any FX 8??? overclocked in any application.

No the G43 suffers from the same problems and is not really suitable for a FX6300, and the G46 would not run a FX8320 well.

Any modern motherboard will run that GPU (and many very old ones too)