How do I extract it from the mainboard of my 2200 alienware 17 laptop.

If you mean the Windows product key, if it's Windows 7 or earlier, look inside the battery compartment.

If it's WIndows 8, no key is supplied -- the Dell install media reads it from the mainboard in the system.

Yeah its out of warrenty. I uninstalled when I got the pc and reinstalled windows to get rid of the bloatware. I didnt need half the stuff that was with it and it was taking up space and slowing it down. Now I need new media to legitimetly get windows 10.

If it's Windows 8, you don't.  You use the Dell recovery media to reload Windows, and no key is required.

If you didn't prepare the Dell recovery media when you received the system, you'll need to get that from Dell -- if the system is under warranty, there's no charge.  If it's out of warranty, there is a charge for it.

You'll need to contact Dell for the recovery media - should cost about $20.