There are two speaker connectors in every SHO (that I've seen, anyway); a square 8-plug connector that goes to the amp, and a flat rectangular 8-pin connector that goes directly to the speakers. You will want to get the amp bypass kit for the Taurus that features this flat 8-pin connector.
Your head unit should be able to power all four corners, but it will not be optimal. Remember, an object that attempts to perform many things at once usually doesn't do any of them very well - just the nature of the beast. It will, however, work just fine for 85% of the people out there.
Make sure you unplug the stock amp's output if you do this; you will blow fuses like a champ if you don't! Having two amps hooked up to each other will result in the weaker one (head unit) blowing fuses... or worse.
I would say unplug everything back there, but if you want to use the stock sub amp then obviously you can't disable the input to the corner amp.
And... therein lies your problem. The sub amp gets its signal directly from the corner amp (silver). If your input signal doesn't go into the corner amp, then... well, I think you can figure it out.
Another problem is that the stock sub amp is (I *think*) 75wRMS@6 ohms. This will get you about 90wRMS@4 ohms. In a perfect world, it'd be 100w, but the world ain't perfect.
This is not enough for most 10" subs. Yes, it will drive it, but it won't be very loud... and, you must remember, that amp is - at the very least - 13 years old. Electronics degrade over time and the signal (and sound) will degrade as a result.
Any modestly decent head unit will have at least one set of pre-outs. Do yourself a favor and buy a nice, decent sub amp (the Alpine MRP-T350 comes to mind [it's what I have

]) that leaves you room for expansion. I say this because you -will- want more. It's human nature. I started out with one JL10W0, and that just wasn't enough. I have 2 10W1s now and that's enough for me - for now - but my wife has 2 10W3s, and they pound like the dickens.
But I digress. You can run the stock sub amp without the stock corner amp... but not easily. It's easier... and, in the end, more rewarding, if you get a real sub amp.