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While it has played out in Ohio and Florida, the backlash against Republican governors began in Wisconsin, where Gov. Scott Walker was the first to take on public unions. He and some of his legislative allies now face recall elections over that fight.

In Michigan, support for newly elected Gov. Rick Snyder fell after he unveiled a budget with deep cuts to education spending while lowering taxes on businesses.

Michigan and Wisconsin handed electoral votes to Mr. Obama in 2008, but Republicans have eyed the states for years, believing voters there could soon swing toward the GOP at the national level.

Democrats, though, said that is not likely to happen in 2012.

Margie Omero, president of Democratic polling firm Momentum Analysis, said many of the new Republican governors jumped so deeply into controversial policy fights that they never got the honeymoon period that voters usually give newly elected politicians. She said that will bleed over into next year’s elections.

“I think it’s going to really have an effect across the board from the top on down and back up again,” she said.