Generally speaking, there's no cooling off period when buying a new or used car. There are a few exceptions but they are pretty narrow and limited.
That said, you still may be able to unwind your deal, but it depends on the circumstances. For example, if the car was sold "as is" and the dealer had no reason to know or suspect that the car had defects, they may not be responsible to fix it.
Carfax collects information from a wide variety of sources, but there's stuff they miss and there's a time lag in getting the information collected and published. It's a good resource but it's not foolproof. Carfax has a claim procedure you can use in case you buy a used car with DMV-reported incidents, but that would depend on what sort of defect you have run into.
I think your best bet would be to contact your local consumer protection office and run the facts by them. Actually your best bet may be first talking with the dealer and trying to work something out - see the unwinding the deal links in the What's the cooling off period to back out of my car purchase? guide.
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