Is the HP Pavilion P6720F a good gaming computer?

No, in two ways. One- HP is made with cheap parts. I own a computer repair shop and I see HP's more than any other brand. Two- If you want good gaming, don't get an integrated video card; it will look AWFUL. A dedicated video card doesn't share with the processor like a integrated one does. If you want a good gaming computer, check out Asus. Don't get the EEE PC though. It's a netbook and those aren't good for... anything.