Relevant specifics: I have an Asus M4A785T-M motherboard with an integrated Radeon HD 4200, and a Sapphire Radeon 5750 graphics card. OS is Windows 7 64-bit, and I have a dual-screen setup.
I have never dealt in SLI/CrossFire before, but would like to get these two graphics chips to wok together. I'm told it should be possible somehow, but cannot find out how.

Dunno about SLI, but I think that with Ati cards in Crossfire you must have two identical non-internal graphic cards, and a motherboard that supports it.
EDIT:
This might help you more:
http://ati.amd.com/technology/crossfire/tutorials.html

Crossfire, yes, that's two compatible (not necessarily identical) cards strung together.
Hybrid crossfire, on the other hand, makes the onboard graphics and the extra card work together.
A similar tech may be available with nVidia's SLI, though I just switched back to ATI.

Hmm, both the chipset and the card may be a tad too new to have any info for them. It's mostly a fun why-not case though I may have bought another motherboard if I had known earlier that they couldn't work together (like one with more card slots or SATA ports instead).
Ah well, a latest-gen DX11 card is much better than my previous GeForce 7950 in any case. Not to mention the new AM3 3GHz Athlon II X2 and 8GB of DDR3 to go with it, replacing the old S939 2GHz Athlon X2 and 2GB or DDR1.
Rep to both for finding some info at least.