Re: How can you recover a CD key?

You need to contact the EA Game Advisors at: http://help.ea.com/contact-us.

They are the only ones who can check your account (on the back-end) and confirm that you've got the game registered.  That allows them to authorize it on your EA/Origin account so you can download and install it. 

AFAIK they will NOT give you your original CD-Key.  But once it's installed, you can then see the product code by using the configuration utility from the splash screen.