No onboard vidio but will work with vidio card

The issue you are describing may be with your video card rather than the monitors. Do you have an onboard video jack? If so, hook one of the monitors to this jack and disable the video driver for your installed vidio card. If it works, you have the answer. If you do not have an onboard card perhaps you can borrow a video card from another computer to test if it is indeed your video card or monitors. You might also try to uninstall and then reinstall the driver from your video card maker. They usually have driver downloads on the site. Lastly, you can buy a cheap video card for testing and if you discover it is your video card buy a new one.

If your using a onboard VGA, probably the motherboard's broken.
If your using a PCI VGA, try removing it and put your monitor @ the onboard graphics card.
Good luck.

Although I'm not familiar with that motherboard the usual way to disable the onboard sound is;
Enter the bios by pressing the Delete key or F2 when the machine starts to boot.
Go to the Integrated Peripherals section and find the On-Board line and change it to disable.

Try a pci video card first, computer is so old that any upgrade at all is going to cost you a lot of money.

Your N9600GT provides two HDMI out puts for dual display.
Even though you are currently only using one HDMI. Switch your monitor cable to the other HDMI out put. See if you now have display.
Also while in the bios allow, Page 2-27 in manual, Allocate IRQ to PCI set to [Yes]. Page 2-24, Internal Graphics Mode Select, [Disable]

Luck!
Mike

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Download the MANUAL for the GATEWAY website to see how to diasable the "onboard" card. 

http://support.gateway.com/s/PC/R/1009153/1009153nv.shtml

MY GUESS IS THE NORTH BRIDGE ON MOTHER BOARD IS BAD, REPLACE MOTHER BOARD.

Ok did you uninstall the old drivers first to prevent a conflict?Uninstall vidio driver and relaod new downloaded drivers from dell or uninstall and reload from disk.RICHARDM69