President Obama speaks to students about his 2013 budget at Northern Virginia Community College on Feb. 13 in Annandale, Va., where he also announced a new $8 billion fund to train two million workers for high-demand industries. (Win McNamee/GETTY IMAGES)

President Obama leads former Massachusetts governor Mitt Romney in Virginia, but voters in the commonwealth are evenly divided on the White House’s major policies, a new Washington Post poll shows.

Obama is ahead of the presumed Republican presidential nominee by 51 percent to 44 percent among registered voters. And Romney does no better against Obama than he did in a Post poll a year ago, despite his emergence as the GOP standard-bearer.

The Democratic president has a key advantage in his bid for re­election: The coalition of Virginians that helped propel him to victory in 2008 — young voters, suburban Washingtonians, women and African Americans — is largely intact. Yet the survey shows that voters in the state are split on Obama’s signature health-care reform law and that they remain deeply pessimistic about the way things are going in the country, creating a potential opening for Romney.

Virginia’s changing electorate and Obama’s 2008 win suggested that the Old Dominion is becoming a reliably swing state. But Republicans have spent the intervening years mounting a rebuttal, winning the governor’s mansion in 2009, three more congressional seats in 2010 and control of the state Senate in 2011. The commonwealth is expected to be a battleground state in November’s presidential election.

Christianne Rutan, a Prince William County Democrat who is African American and voted for Obama four years ago, said the president will receive her support again.

“No question, I would stick with Obama,” said Rutan, 22, a mother of two who is a therapeutic recreational assistant at a nursing home. “I think he’s doing an amazing job. He doesn’t get the credit that he deserves.”

By a whopping 97 percent to 1 percent, Obama thumps Romney among black voters, and he has a wide lead among women: 56 percent to 38 percent. Obama’s up by better than a 2-to-1 ratio among those ages 18 to 29, but he faces a challenge among that group in the coming months. Many of them — 34 percent in this poll — are not registered to vote at their current addresses.

Obama was the first Democrat to capture Virginia in four decades, and this week’s schedule indicates his desire to repeat that performance. He will be in Arlington on Friday to hold a roundtable and deliver remarks on student-loan debt. He will return a day later in the guise of a candidate, holding a rally in Richmond meant to mark the official start of his reelection campaign.

Meanwhile, Romney held an event on small business in Chantilly on Wednesday, took in more than $2 million at a Pentagon City fundraiser Wednesday night and made another jobs-focused stop in Portsmouth on Thursday, part of his broader argument that Obama’s policies have stifled the economy. The candidate also plans to deliver the commencement address at Liberty University in Lynchburg on May 12.

The state has been a magnet for outside ad spending, both in the presidential contest and in the marquee U.S. Senate race that is likely to feature former governors Timothy M. Kaine (D) and George Allen (R). Many of the attacks on Obama and Kaine have focused on the president’s health-care law. Yet that issue is a wash in the new poll, with supporters and opponents tied at 47 percent among registered voters — making the legislation slightly more popular than it was a year ago.

Virginians are split down the middle on the administration’s major policies overall, but tend to credit Obama with doing a good job. The president has a 53 percent approval rating, while 44 percent disapprove (it’s a narrower 50 to 47 among registered voters). Republicans in Congress fare less well, with 60 percent of respondents saying they were dissatisfied with or angry about the policies of the congressional GOP.

Virginia Beach resident Philip Read, 45, who lost his public relations job eight months ago when his company closed, said he is leaning toward voting for Obama but is not “enthusiastic.” He said he supported Obama in 2008 but thinks the president has not done a good enough job on the economy.

“What happened with Mr. Obama is that he worked to get Obamacare at any cost that he lost sight of the economy,” said Read, who is an independent. “That looms large in people’s minds. He didn’t completely tackle that.”

Read said Romney needs to show what he can do to improve the economy to get his vote.

Candace Swartz, 41, a Republican from Warren County, said she will vote for Romney.

“To be honest, I’m not really great about any politicians right now,” said Swartz, who works as a quality coordinator for Kraft Foods. “But Mitt Romney seems to be the best one between the two. . . . A lot of it’s the economy, a lot of it is just [Obama’s] general policies. I don’t believe in the health care, Obamacare.”

Obama made early investments in the state, and now has 13 offices open and dozens of paid staff members. Virginia Republicans have seven field offices that will be used to boost Romney as well as other party candidates, and the Romney campaign announced Thursday that Sara Craig, Romney’s state director during the Iowa caucuses, will run his Virginia operations.

Romney has not been able to improve his statewide support from a poll a year ago even as his GOP primary opponents have fallen by the wayside. Yet he leads Obama by a huge margin among Republicans, just as Obama holds a major lead among Democrats. The key to the president’s overall edge can be found in the middle of the spectrum. He has an advantage among independents, and a 23-point lead among self-described moderates.

And if Romney hopes to increase his chances in Virginia, choosing Gov. Robert F. McDonnell (R) as his running mate may not do the trick. More than two-thirds of voters say adding McDonnell to the ticket wouldn’t make much difference in their choice, 11 percent say it would make them more likely to back the Republican ticket, and 19 percent say it would push them toward Obama’s side.

As in the past several statewide elections, the outer suburbs of Washington are among the state’s most competitive battlegrounds — Romney and Obama are essentially tied in the ring that includes Loudoun, Prince William and Fauquier counties. They also are neck and neck in and around Richmond. Obama runs up the score with wide leads in the inner D.C. suburbs and the Tidewater area. Romney’s strength is in central and western Virginia.

The state’s voters may want change, but the desired direction is far less clear than it was before the 2008 election. In the summer of 2007, nearly two-thirds said they were dissatisfied with or angry at the George W. Bush administration; far fewer now say so of Obama’s. More now say they’re unhappy with Republicans in Congress than the executive branch. And although 83 percent of voters four years ago said they thought the country was on the “wrong track,” that number now sits at 65 percent.

The poll was conducted by telephone April 28 to May 2 among a random sample of 1,101 Virginia adults, including 964 registered voters and users of both conventional and cellular phones. Results from the full survey have a margin of sampling error of plus or minus 3.5 percentage points.

Polling manager Peyton M. Craighill, polling analyst Scott Clement, and staff writers Amy Gardner, Anita Kumar and Laura Vozzella contributed to this report.