HOUSTON — Raising campaign cash in Republican territory, President Barack Obama on Friday hailed a rebounding economy and accused Republicans of banking on voters having “amnesia” about the steps that led to a brutal economic collapse.

“The recovery is accelerating. America is coming back,” Obama told 600 supporters at a Texas fundraiser.

Bidding for re-election, Obama bounded between a rally-style event in a sprawling Rolls-Royce manufacturing plant south of Richmond, Va., to a pair of Houston fundraisers. Framing the trip: a new monthly jobs report showing employers 227,000 jobs in February, the latest sign that the economy is headed in the right direction.

Every month’s jobs report is seen as a barometer of the economy and an important factor in the presidential race. The unemployment rate held steady at 8.3 percent, the result of more Americans looking for work as job growth takes hold month by month.

The jobs report and split loyalties among Republican voters assessing Mitt Romney, Rick Santorum and the rest of the GOP field give Obama’s team renewed confidence that the path he has forged could help him win re-election and rebuild the economy. But privately, his advisers know that outside factors in the United States and abroad — from high gasoline prices to instability in the Middle East — could still derail his political and economic ambitions in the months leading to the fall election.

Pointing at Republicans, Obama said: “They think you have amnesia. They think you’ve forgotten how we got into this mess.”

Defending his health care law, Obama said preventive care for women was now covered, including “checkups, mammograms, birth control. We fought for this because the top doctors and medical experts in the country said this kind of preventive care saves women’s lives,” he said, arguing too that it saves money.