Republican presidential nominee Donald Trump, left, and Democratic presidential nominee Hillary Clinton meet for their first debate on Monday at Hofstra University in Hempstead, N.Y. (Melina Mara/The Washington Post)

With just five weeks of campaigning left, Hillary Clinton and Donald Trump are scrambling to win over female voters in America’s suburbs as well-educated white women have emerged as perhaps the presidential campaign’s most pivotal swing voting group.

The Democratic and Republican nominees and their surrogates are making direct appeals to female voters in campaign appearances in the suburbs of North Carolina, Ohio, Pennsylvania and other battleground states; in television ads on channels such as Bravo and shows such as “Dancing With the Stars”; and on Facebook, where people’s feeds inundate them with campaign commentary.

But polls and focus groups, as well as interviews with women here in the battleground state of New Hampshire, show that Trump — who spent the past week haranguing a Latina former beauty queen over her weight — has a considerable challenge with many female voters.

“It’s very clear that Trump is doing extremely well among white non-college-educated men . . . but white women with a college degree is a huge impediment to getting where he needs to be,” said Whit Ayres, a veteran Republican pollster. “I’m not sure what he can do about it given all the comments he’s made about women over the last 15 months.”

Trump’s troubles were evident here in Nashua, a commuter exurb of Boston, where six women in a knitting circle were lounging on the couches and armchairs of a yarn shop the other day talking about — what else? — Trump. They were Republicans, Democrats and independents, all of them moms — and all of them ready to give him a permanent timeout.

Florida voters listen to Democratic presidential nominee Hillary Clinton during a Sept. 6 campaign rally in Tampa. (Melina Mara/The Washington Post)

“You just want to smack him,” said Pam Harrison, 56, who voted for Republican Mitt Romney four years ago.

Watching Trump debate reminded Kristen Schwartz, 40, of dinner-table conversations with her in-laws: “It’s not polite to interrupt people, but if you stop to breathe or think about your point, they just talk over you and the conversation just gets louder and louder and louder.”

In an unnerving campaign season, what keeps their anxieties in check is the belief of Sandy Zielie, 46, the shop’s owner: “Women are going to save this country this election.”

Female voters may not save the country in the way the knitters of Nashua would like, but they almost certainly will swing the election. Clinton and Trump are targeting many intersecting groups and subgroups of swing voters, but strategists for both campaigns said white women with college degrees are at the top of their lists.

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Trump’s temperament has been a flash point since he entered the race, especially among women. So it was that many of Clinton’s surgical strikes against Trump in their first debate were designed to sow fresh doubts, especially when Clinton recounted how he had shamed Venezuelan-born Alicia Machado, the 1996 Miss Universe pageant winner, for gaining weight. The controversy continued for days, as Trump lashed out at Machado, including maligning her in an erratic series of tweets starting at 3:20 a.m. Friday.

The Miss Universe episode has not gone over well with the female voters he needs to win over.

The Sept. 22 Washington Post/ABC News poll is packed with data. Here are a few of its most interesting findings. (Peter Stevenson/The Washington Post; Photo: Jabin Botsford, Melina Mara/ The Post, AP)

“I have always voted Republican, but I don’t feel like I could vote for Trump this year,” Rosanna Koehlert, 58, a college graduate and housewife, said as she shopped the other day in Merrimack, N.H. “He shouldn’t be making fun of people and making them self-conscious about the way they look. That’s not what a president should be.”

The modest lead Clinton has held in national and state polls can be attributed to her outsized advantage among white women with college degrees. Four years ago, Romney carried this demographic over President Obama, 52 percent to 46 percent, according to exit polls. Yet Trump is losing it badly — 32 percent to Clinton’s 57 percent in a late September Washington Post-ABC News poll of likely voters.

Among white women without college degrees, however, Trump leads Clinton 52 percent to 40 percent. The two candidates are virtually tied among white women overall: 46 percent for Clinton and 44 percent for Trump, according to the Post-ABC survey.

For Clinton, who would make history as the nation’s first female president, winning a larger share of white women voters than Obama did could help her offset any relative erosion in support from young voters, for instance. It also could balance any rise in turnout among white men for Trump, a group with which he enjoys an over 2-to-1 lead.

“Everything’s about swing voters,” said a senior Clinton campaign official, who spoke on the condition of anonymity to discuss campaign strategy. “We know that white suburban women are critical for both parties . . . and the lowest hanging fruit for expansion among that group is more likely to be college-educated white women.”

Meanwhile, Trump is trying to squeeze as many votes as he can from whites generally — including a subgroup with which he is quite unpopular, college-educated women — to offset losses among blacks, Latinos and other minorities.

“Some women have already gone into their respective corners, and they will support Clinton or Trump and not move,” said Kellyanne Conway, Trump’s campaign manager and pollster. “But the persuadables are swinging back and forth because they know there’s always more to learn, more to see, more to know.”

Margie Omero, a Democratic pollster who closely studies the political attitudes of female voters, said women have been cool to Trump’s candidacy for several reasons.

“It’s not just about the toxic language he uses towards all the various women he knows publicly, but it’s also his lack of fluency on policy, his lack of understanding about caregiving,” Omero said. “It’s clearly not his comfort zone. He’s got this very harsh tone, in general, that a lot of women respond very badly to.”

For months now, Clinton’s advertisements have used Trump’s own words to undermine his character. The spots have been aimed at women and evoke visceral responses. One of the campaign’s most recent ads, called “Mirrors,” depicts girls looking at their reflections as Trump is heard talking crudely about women’s bodies.

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Clinton is laboring to persuade voters who identify as “moderate,” regardless of their party affiliation. In 2012, moderates made up 41 percent of the electorate, and Obama won them 56 percent to Romney’s 41 percent, exit polls show. Many moderates live in suburban areas, and although they cross over into many demographic groups, the Clinton campaign’s private data suggest they hold similar values and beliefs.

“There are threads here that make these groups very important to winning the presidency,” said the Clinton campaign official, saying they support same-sex marriage, prioritize climate change and education, and recoil against candidates seen as sexist or bigoted.

To repair his image among moderates, Trump has enlisted his elder daughter, who rates as one of his most effective surrogates according to the campaign’s internal research. Ivanka, 34 — who has three small children and is an entrepreneur and an executive in her father’s real estate company — helped Trump craft a child-care policy, which they rolled out together last month in a Philadelphia suburb.

Ivanka stars in a new ad, her first of the general election, in which she says that her father “understands the needs of a modern workforce” and is committed to changing “outdated labor laws” to support women with children. The spot, called “Motherhood,” will air this week nationally and in swing states on women-focused cable channels, including Lifetime, Bravo and OWN (the Oprah Winfrey Network), as well as on network prime-time shows including “Dancing With the Stars,” “Grey’s Anatomy” and “The Voice.”

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Trump’s women-focused ad campaign will continue with a second spot, called “Childcare,” that casts Trump as a champion of affordability and fairness for middle-class families, Conway said.

This is partly an attempt to change the subject from Trump’s temperament and character, and to tell voters that he is more of a change agent than Clinton.

“If this election is about Hillary Clinton and is fought on the issues, Donald Trump wins,” Conway said. She ticked through the issues she wants Trump to focus on in the final weeks: security (not only terrorism, but also opioid abuse), affordability, fairness and ethics.

Conway said the Trump campaign recently analyzed the advertising of Clinton and her super-PAC allies and found that the two issues discussed the least were education and health care.

“I said, ‘Bingo! That’s what we’ll talk about every day — health care and education,’” Conway said. “She says she’s been ‘fighting for women and children,’ but where’s the product? Where’s the deliverable? Why do so many women live in poverty? Why do so many women not have health insurance?”

It may be difficult for female voters to tune everything else out, however. Lisa Faust, 45, one of the knitters here in Nashua, was trained as a chemical engineer and is raising a son with Down syndrome. She can’t stop thinking about the time Trump mocked a disabled journalist.

“What’s distressing to me is that Trump has made it socially acceptable to embrace bigotry and racism,” Faust said. “How do you say to an 11-year-old child, ‘That’s not acceptable behavior,’ but the leader of our country thinks it’s acceptable? That really gets to me.”

Emily Guskin in Washington contributed to this report.