President Obama could really solidify his position in history as a progressive President by working with the Republican House to pass their ideas.
If Obama can work with Repubicans to implement the defitcit commission recommendations, and extend the Bush Tax Cuts then the Right could no longer demonize him with credibility.
If he cuts the long-term debt problem with $3 spending cuts to $1 tax increases on the backs of their votes, he would gain instant cache among moderates and independents.
While the left would be very upset, the President would be able to argue:
That he passed healthcare legislation that extended healthcare to tens of millions, curbed insurance company abuses, and improved the fiscal outlook of the sector.
That he revamped the student loan industry to take away bank subsidies.
That he reigned in Wall Street and put in rules to prevent another economic crisis.
...and on and on.
So the upshot of working with the Republicans is that they have to accept what he's already done. If the Republicans on Wall Street like a grand budget deal, and the lower taxes, they will begin to create more jobs and improve the economy.
The anger on the right against the President can't last in a good economy.
So the President can get re-elected by ceding some ground while holding the fort.
If Obama gets re-elected with healthcare in tact, especially if he cuts the deficit and shores up entitlements, he'll go down as one of the greatest Presidents ever.