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Mr. Greenberg’s poll shows that Republicans have an uphill battle ahead.

Among likely voters, 53 percent of respondents said they approve of the way Mr. Obama is doing his job, while 25 percent said they approve of the way Republicans are running the House.

On the issues of gay rights and tax rates for millionaires and corporations, more than 6 in 10 of the respondents said the Republican Party was “growing extreme and out of touch.”

More than 5 in 10 said the party was missing the mark in dealing with Wall Street regulations and climate change, as well as aid to the poor, immigration and women’s issues.

“There are a very large number of issues where they are seen as extreme,” Mr. Greenberg said in a conference call with reporters. He said the party scores best on how it has dealt with assault weapons and gun violence.

The poll of more than 1,800 voters and likely voters was conducted Jan. 10-14 and had a margin of error of 3.2 percent.

Democrats have had a field day with Republican struggles.

Rep. Sander M. Levin said this week that in his 30-plus years on the tax-writing House Ways and Means Committee he has never seen such a radicalization of the Republican conference, and blamed gridlock in Congress on the tension within the party ranks.

“I think that the results of that were shown in the difficulty that the speaker had in the last few weeks,” Mr. Levin said.

He added that he thought “in terms of mainstream America, that the Republican ranks have changed much more dramatically than … the texture of the Democratic caucus.”

Meanwhile, the Democratic Congressional Campaign Committee also blasted out a tongue-in-cheek email offering suggested retreat agenda items the Republican Party might want to consider to help “explain their extremism and dysfunction to the American people.”

The committee’s recommendations included: stop talking about “legitimate rape,” study Science 101 and practice interacting with women and minority voters.