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I am assuming that the correct way to set this up would be to get a sound card that outputs 5.1 channels and run one cable
There's no real correct way, but what you described should work. Althoguh you will want to ensure you have some form of bass management in your PC to direct bass to your sub output.
An easier route is to simply ensure your new 4 channel amp has passthrough RCAs, and then conenct those to your sub amp, thus reducing the need for a new soundcard.