This is an AMD based system, you said quad core, I say octo-core!
Here you are, the total is $366.96, excluding shipping of course.
Western Digital 7200RPM S-ATA 6 GB/s, 6$MB Cache 1TB Hard Drive. (You could get a 500GB HDD and a 64GB SSD, if you wanted)
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Corsair Vengeance Pro 8GB RAM Kit (2x4GB) DDR3 - 1866MHz
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AMD FX-8120 Octo-Core processor, you could cheapen out and get a quad core, like you asked but this is still in budget!
This is a bundle with a GIGABYTE GA-970A-DS3 motherboard, it has all the stuff you need for upgrading in the future, whether you want a graphics
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I think you might need a graphics card with this... I don't think there is an iGPU, although you can buy a nVidia GT630 or something if you aren't gaming.