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“Speaking to independents, conservative Democrats, moderate Republicans — people who know we have to do this — why should the Democratic Party be trusted? And are we willing to make some of the tough decisions to actually head this country in a better direction?” he said.

Mr. Obama replied with a stinging attack on Republicans’ failures.

Mr. Bayh has always cut a unique figure on Capitol Hill, and has taken to wearing sneakers with his business suits. That drew a quip from the president at Senate Democrats’ issues conference, to which Mr. Bayh replied, “You’ve got to stay light on your feet around here, right?”