It honestly doesn't matter what Obama does in the next two years. The
Republicans already know the only hold part of a hand... they can't push their agenda with just the House.
Oh yeah! Right! The GOP didn't take over the world this month... only our House of Representatives!! Obama has his veto. The Senate is still a morass for legislation. Nothing's going through there. Of course the new talking point will be that Obama isn't 'cooperating'. If nothing gets done, and it won't, then Obama will be the hyper-partisan.
The next two years really comes down to framing your argument. That and a race to show the people that you're doing the best for them. Obama will be running around building cred by adding jobs, to push for tax cuts for the middle class only, to expand on the health care bill through explaining what is going into effect,etc.
The GOP on the other hand will take the business route. They will give every cut and expansion they can for Big Oil and Wall Street. Sure! Go ahead and screw the environment again, we're in charge! And along the way keep shoving bills that looks like they are for the people but are probably 66% filled with giveaways to their friends.
Meanwhile, the Teabaggers will become the
biggest pain in the ass in government. They will start going off message with the GOP. They will make promises they can't keep as well as expect harsh new ways that their conservative overlords won't want. At some point they will tip over and the GOP will start pushing them back out. Assuming of course they don't need the seats they hold or they can swap them out for main-line folks.