The National Rifle Association provoked a furious response from the White House on Wednesday by releasing a video accusing President Obama of being an “elitist” and a “hypocrite” because he opposes posting armed guards at schools, while his daughters have Secret Service protection.

The video also raised, for some people, questions about whether the gun rights organization might have been too strident, even for its own members.

The White House lashed out at the N.R.A. even as Mr. Obama stood with young children to unveil broad proposals to create tougher gun laws and use the power of the presidency to keep guns out of the hands of criminals.

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