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Mr. Obama, 51, a former U.S. senator who makes his home in Chicago, made history in 2008 by becoming the first black man to win the White House. His inauguration drew a giant crowd, and he took office amid hope that he could unite a fractured country after President Bush and right an economy reeling from the Wall Street collapse just months before his election.

He had major legislative accomplishments during his first two years: With huge majorities in Congress, Democrats passed the economic stimulus, new financial market regulations and the health care reform that has become Mr. Obama’s signature achievement.

But a voter backlash in 2010, which Mr. Obama deemed a “shellacking,” gave the GOP control of the House, curbed Democrats’ majority in the Senate and left conservatives pushing for deep spending cuts to try to contain the red ink that spilled across government balance sheets.

The country ran trillion-dollar deficits all four years Mr. Obama was in office, and the overall national debt topped $16 trillion.

Amid that playing field, Mr. Romney, 65, a former one-term governor of Massachusetts who made his personal fortune at the helm of Bain Capital, vowed to bring bipartisanship and a businessman’s can-do know-how to Washington.

He also promised big ideas, including an overhaul of Medicare and Medicaid to try to cap the rising costs of entitlements, which threaten to crowd out all other federal spending in coming decades.

Mr. Romney underscored his commitment by picking as his running mate Rep. Paul Ryan, chairman of the House Budget Committee, whose budget plans have deeply divided Congress over the past two years.

But Mr. Ryan at least was able to say that his fellow Republicans in the House passed a budget.

Not so for the Senate, where Democrats have failed for three straight years to pass a blueprint for spending.

Passing a budget is just one of the challenges awaiting the election victors.

The next Congress will be sworn in on Jan. 3, and Mr. Obama retakes the oath of office Jan. 20.

The government is poised to run another trillion-dollar deficit in 2013, and the gross federal debt has topped $16 trillion. Meanwhile, the unemployment rate remains stubbornly high, but Washington has left few tools untried in its push to spur job creation.

Some work won’t even wait for the new year, however.

Congress returns next week for a lame-duck session to grapple with the Bush-era tax cuts, which expire Jan. 1, and/or the $110 billion in automatic spending cuts, which are due to take effect Jan. 2.

With the election so narrowly decided, the winner was unlikely to carry much of a mandate into the next term. Combined with a split Congress, that meant there was little chance of easy answers to the items stacked up on the agenda.