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“One thing they could do to dig their way out of this is to get a bipartisan agreement on something big — entitlement reform, or tax reform that doesn’t increase taxes,” said John Feehery, president of communications at Quinn Gillespie and Associates in Washington. “He’s got to really try to go into deal mode and get some accomplishments that can dig him out of this morass.”

Mr. Feehery, who was a spokesman for House Speaker J. Dennis Hastert, Illinois Republican, also predicted that congressional Republicans will help Mr. Obama unwittingly by overreaching in their investigations of the administration.

“That’s what always happens, especially with Republicans,” he said.

Whether or not Republicans turn overzealous, there is some evidence that their hearing on Benghazi last week caught the public’s attention. The Center for Media and Public Affairs, a nonpartisan group in Washington, analyzed news coverage in the months after the Sept. 11 attack and compared it with news reports in two weeks leading up to the House hearing. The group said the Republicans’ efforts brought “more balance” to the story.

The center found that since House Republicans released a preliminary report on Benghazi on April 24, 58 percent of sources in major newspapers have faulted the Obama administration’s response to the attacks. In the months immediately after the attack, the analysis said, the news media were four times more likely to report the administration’s discredited claims that the Benghazi assault grew out of a spontaneous protest over an anti-Islam video rather than a terrorist attack.

Mr. Feehery said the administration’s loss of control over its message on the president’s agenda is typical for second-term presidents rapidly confronting lame-duck status.

“You get a real arrogance of power,” he said. “He thinks he’s got power, but every day their power diminishes. I think their guard gets down.”

Mr. Feehery thinks there is worse to come for the administration as it tries to implement the president’s health care law in the wake of the IRS controversy.

“That’s the real scandal — big government,” he said. “It’s the idea that people don’t trust their government. You have the IRS doing these things, and then you’re going to have all these agencies basically breaking the news to people that their health insurance premiums are going to increase by 300, 400 percent. That’s going to make [Mr. Obama] very, very unpopular.”