For Bill Bradley, the free ride of being a long-shot in the presidential sweepstakes is over.

Now that the former basketball star has established himself as a real threat to Vice President Gore's political future, Bradley is facing sharper questions from voters on the campaign trail and a whole new set of challenges befitting a serious bid for the country's highest office:

How does he expand his base beyond those Democrats who despise the Clinton administration? How does he capture the loyalties of Democratic voters inclined to support the man groomed for the job? How does he turn himself and his campaign into the kind of cause that motivates his party's voters to jump on board and cast Gore overboard?

And perhaps most significant, how can Bradley run to the left of Gore to capture the nomination--as he has signaled he intends to do by broaching such issues as universal health care and tougher restrictions on handguns--in a party that has shifted to the center, and especially after 18 years in the Senate that marked him as a cautious moderate?

As he confronts such questions, Bradley has the advantage of his longtime celebrity status, which gives him a certain cachet among voters of all ideological persuasions.

Last week, in his formal kickoff in his home town in Missouri and on the campaign trail in Iowa, Bradley gave some sense of the strategy he plans to use to propel himself past Gore. He intends to present himself as a candidate restoring the tradition of idealism and battling injustice exemplified by the late Sen. Robert F. Kennedy. He plans to recast himself as a risk-taking, liberal-minded activist who would use the muscle of government to solve such big national problems as child poverty.

Perhaps most significant, he will try to fuse voters of all ideological stripes by offering himself as a candidate of rectitude and integrity, which he expects will be welcomed after eight years of controversy, investigations and scandals surrounding the Clinton administration.

So far, Bradley has built a credible campaign organization and has raised more than enough money to be competitive with Gore. But in national polls, Bradley has yet to make a breakthrough: In the most recent Washington Post/ABC News survey, Gore held a decisive 69 percent to 24 percent advantage.

In addition to his underdog status, Bradley faces additional hurdles: Most of his support comes from anti-Clinton voters and does not appear to be based on his specific appeals. He has, for example, gone out of his way to stress his commitment to racial reconciliation, but African American voters remain one of Gore's strongest bases, backing him by 85 percent to 15 percent for Bradley, and show little signs of wavering.

Some political analysts question whether Bradley's approach will succeed. "Bradley has to find some way to answer the question why--if he feels so passionately about these things, gun registration and health care--why didn't he do them before?" said David Axelrod, a Chicago Democratic consultant who supports Gore but is not on the campaign payroll.

Bradley's presentation of himself as a man of exceptional moral rectitude also could prove difficult for a politician who served three terms in the Senate, Axelrod said. " 'I am worthier and I am better and I stand apart.' That is a hard message to sell in politics, especially for someone who's been in it," he said.

National polls are poor reflectors of an insurgent's prospects at this stage, five months before any votes will be cast. Bradley is depending on making a breakthrough in either the Iowa caucus in January 2000 or the February primary in New Hampshire. At the moment, New Hampshire is Bradley's best state, with two polls showing him within 4 to 7 percentage points of Gore.

As he prepares for those critical states, Bradley faces the cold reality that voters who now recognize his political viability are demanding that he make the kind of hard choices that, while pleasing current supporters could anger other voters.

This last challenge became apparent last week at what was supposed to be a low-key meeting with 15 likely supporters that turned into a hard-nosed question-and-answer session.

Jamie Swain, whose husband is on the school board here, was first to take the floor: "Is it true that you are in support of school vouchers, and, if so, why?" she asked, raising an issue of vital importance to a major Democratic constituency, the National Education Association and its Iowa affiliate.

"No, I am not in support of school vouchers as the answer to the problems of public education," Bradley declared, taking a position that appeared to be fully in line with the NEA.

"When I was in the Senate, I voted for vouchers as an experiment in six urban areas of the country," Bradley said. After years of advising overworked parents to spend their spare time trying to improve their local schools, he "thought, well, why don't we try and experiment." But, he quickly declared, "I don't think vouchers is the answer to the national school problem."

Bryan Von Kley then hurled a fastball: Title IX of the civil rights laws requiring equal sports opportunities at colleges for men and women had turned into a confiscatory quota system in which colleges are dropping such sports as wrestling, golf and track and field instead of adding new women's sports. He pointed out that Princeton, Bradley's alma mater, recently dropped wrestling.

Bradley, appearing increasingly uncomfortable, said Title IX "should work by expanding the opportunities for women, not curtailing the opportunities for men." When Von Kley pursued the topic, Bradley said that if he "had a choice of well, let's go back on Title IX or have the occasional university that does it the other [wrong] way, I'd say, let's not go back on Title IX."

But anecdotal evidence, based on the kind of supporters Bradley attracts, suggests that some themes, including renewing the liberal commitment to fight poverty and end racism, are paying some dividends.

Shannon Anderson, a 20-year-old political science major at Grinnell College, said she volunteered to work for Bradley because he is willing to take stands on race and the environment, in contrast to Gore, who she said is "trying really hard to be moderate, like he doesn't want to offend anyone. It's really hard for him to break away. I've heard Gore speak, and he's a really nice guy, but he just doesn't take the stands Bradley is willing to."

CAPTION: Democratic presidential candidate Bill Bradley talks to Vivian Nelson of Nevada at party event in Ames, Iowa.