I have had this problem on my Asus G51j Intel I5, Nvidia GTS360M graphics card. Just updated the drivers to latest off Nvidias site.

Most games I play, If I hook it up through an external monitor(this happens whether it's HDMI or VGA). The laptop hard shuts down after about 20 minutes, give or take. This tends to happen on more graphically intense games like Call of dutys, bioshock.

I'm running Windows 7, latest nvidia drivers. Games work fine on my laptops screen, and Movies, general computing work fine on the external displays.

The laptop always sits on a cooling pad and the latest time was in front of my AC unit, I ran heatfan or whatever the heat tracking software was, and my temperatures were normal through a shut down.

This has been happening for the life of the laptop. I dont play very many games, and even fewer on an external monitor, so this issue doesnt come up much

Is it possible I have a faulty graphics card? Is there anything else I can try?