Did you install a retail version of OS? If you did, then the disc should have come with a product key.
I think it's better for you to just buy the recovery disc because if you will use the recovery disc, it will not ask for a product key anymore. It is activated already.
Even if you can find the product key underneath the laptop, it will still not work on your retail version of OS because that product key underneath the laptop is the product key for the preinstalled OS.