On top of the above excellent advice, I will add:

CarFax is a tool, they overly promote themselves, and they are not as good as advertised. They also don't stand behind their guarantee unless you have real good evidence, you've covered all the fine print, and you have a good lawyer. There are also other companies besides them.
http://www.consumeraffairs.com/automotive/carfax_inacc.html

And from: http://public.findlaw.com/congen/caw_automobiles_buying_used.htm
The websites www.carfax.com and www.autocheck.com sell information on the history of vehicles gathered from state motor vehicle departments and other sources. These reports are helpful but incomplete-they do not guarantee that a vehicle is accident-free.