I have an hp puvillian with A10-5700 integrated graphics; I just got a new AMD FireGL 3100 and installed it. When I started my computer the only monitors that worked were the ones that were plugged into the 3100. Is there any way a external GPU and integrated graphics can work together?
Both of those are internal, just clarifying for you. The Integrated graphics are disabled unless your CPU supports AMD Dual Graphics which is essentially Crossfire between the integrated graphics on the CPU and a compatible graphics card.
The 6670 is supported by Dual graphics with your APU. The FireGL is not. Dual graphics works fine but it is far more cost effective to spend a little bit more money on a better GPU because both the FireGL 3100 and the 6670 are pretty much paperweights. The A10 you have is a good CPU though so I'd suggest getting your hands on an R7 260 or 750ti. Anything less isn't worth the money.