How can I find my model #?

My MacBook Pro mid-2010 15" is having major problems. Screen sometimes flickers and flashes in green and purple. Sometimes won't start. It was pronounced dead at one Genius Bar. Another told me that the battery needs to be replaced. I'd like to try replacing the battery, but I cannot find the model #. The bottom of the computer only gives the serial number; no other text.

Any suggestions here?

I'm not sure it's even worth it to buy the battery. Will that really rejuvenate the computer, or should I save the money for a new laptop?

Thanks for your help!