MANCHESTER, N.H. - Republican Mitt Romney launched his general election campaign against US President Barack Obama after sweeping five primaries on Tuesday, condemning the president for false promises and weak leadership and declaring "a better America begins tonight."Romney claimed victory in the Republican nominating race and outlined the themes of his campaign against Obama, asking Americans if they were better off under his administration and accusing the president of failing to deliver on his campaign pledges of hope and change.Romney's speech, focused almost exclusively on the economy, came after he rolled to easy double-digit victories in five Northeastern states - Connecticut, Delaware, New York, Pennsylvania and Rhode Island - with a combined 231 delegates.The wins moved Romney, who entered the night with 695 delegates, closer to the 1,144 he needs to formally clinch the nomination, a milestone that is still weeks away.Romney effectively ended the Republican race on April 10 when his top rival, Rick Santorum, suspended his White House campaign, but the speech in the general election battleground state of New Hampshire was the first time he had claimed the mantle of presumptive nominee."After 43 primaries and caucuses, many long days and more than a few long nights, I can say with confidence, and gratitude, that you have given me a great honor and solemn responsibility," Romney said.Gingrich yet to concede Republican nomination to RomneyThe primaries on Tuesday could be the beginning of the end for another remaining rival, former House of Representatives Speaker Newt Gingrich. He had said he would reassess his candidacy if he did not win the primary in Delaware, where he had campaigned heavily in recent weeks.Romney beat Gingrich by more than 30 percentage points in Delaware and won all of the state's 17 delegates. But Gingrich did not concede during a speech in North Carolina after the vote. He said he planned to continue his schedule in the state this week.Gingrich has won only two primary contests - in South Carolina and in Georgia, which he represented in Congress - and his campaign is deep in debt, but he has been hanging on to keep pressing Romney on conservative issues.The other remaining candidate, libertarian Texas Congressman Ron Paul, said again on Monday that he would not drop out of the race even after Romney clinches the nomination.Romney, a former governor of Massachusetts, enters the election campaign bruised from a bitter primary battle with a series of challengers who questioned the sincerity of his conservative views.He is faced with the task of consolidating support from conservatives who distrust him for the more moderate positions he took as governor of liberal Massachusetts, particularly his support for a healthcare overhaul that became a precursor to Obama's federal plan.At the same time, he must turn toward winning over undecided independent voters who are likely to decide the election, and bolster his lagging support among women, Hispanics and young people.Romney is also launching a search for a vice-presidential running mate. Included on his long list of potential No. 2s is Florida Senator Marco Rubio, who campaigned with him in Pennsylvania on Monday.In spite of those challenges, Romney opens the election campaign in a relatively strong position, with national opinion polls showing a tight race with Obama.Obama, who has the advantages of incumbency, is well-liked by Americans but there are deep doubts about his handling of the economy and anger over high gasoline prices that may, however, be trending downward.He must convince Americans that his prescriptions for the sluggish economy will lead not just to a stock market gains, but to real job growth among the dispirited middle class.Obama visited two battleground states on Tuesday, North Carolina and Colorado, to appeal for support from young voters with a call to make education more affordable.He stressed his modest background and the student loans he needed for college, references that seemed designed as a swipe at the multimillionaire Romney.Given the stakes, the presidential campaign will likely be negative as the two sides battle through TV and radio ads. Both candidates and the outside groups that support them are building campaign accounts likely to reach into the hundreds of millions of dollars.Their goal is to portray the other side as being out of touch with the concerns of ordinary Americans and unable to solve intractable problems, from debt to deficits, to caring for an aging population."What do we have to show for three-and-a-half years of President Obama?" Romney asked during a speech in New Hampshire. "Is it easier to make ends meet? Is it easier to sell your home or buy a new one? Have you saved what you needed for retirement? Are you making more in your job?"Romney, a former head of a private equity firm who has put the struggling economy at the center of his campaign, slammed Obama's economic policies and said they had led to "hopes and dreams diminished." He promised he would offer a better chance to those who are struggling."People are hurting in America, and we know that something is wrong, terribly wrong with the direction of the country," he said."This election is about the kind of America we will live in," he said. "In the America I see, character and choices matter. And education, hard work, and living within our means are valued and rewarded."