Is this the motherboard you are currently using, right?
http://h10025.www1.hp.com/ewfrf/wc/document?docname=c03117539&cc=us&dlc=en&lc=en#N87
Seems like your motherboard form factor size is a micro ATX which is a good sign, at least it means that the computer case isn't using any special form factor limited to that manufacturer.
It's probably best to get another micro ATX board. The 5770 card should be able to crossfire.
The only other issue is that you will have to upgrade the PSU. The PSU that comes in your computer appears to be a 460W OEM PSU, correct?
You should probably go for a 500-550W good quality brand PSU such as from EVGA, XFX, Seasonic, or even Corsair possibly.
I ran into an issue since there is almost really no motherboard that is mATX and is an AMD motherboard for FX processors (AM3+) that has two PCIe x16 lanes. I'm not sure if you can fit a full size ATX board in your desktop though but if you can take pictures of the inside of your computer, that would be very helpful.
You're probably better off getting a single powerful GPU.