House Majority Leader Eric Cantor (R-Va.) speaks at the American Enterprise Institute, on Feb. 5. (Mark Wilson/GETTY IMAGES)

After years in which the Republicans’ answer to almost every question was to reduce spending and lower taxes, some of the party’s most influential voices are beginning to suggest that it is time to take a broader — and softer — focus.

The latest to join that movement is House Majority Leader Eric Cantor (Va.), who in a speech on Tuesday called for Republicans to “focus our attention really on what lies beyond the fiscal debates” and to create “conditions for health, happiness and prosperity.”

As is fashionable these days, Cantor even came up with a catchy Twitter hashtag for his rebranding effort: #MakingLifeWork.

Changing the subject from fiscal issues could prove difficult, however, given that congressional Republicans are squaring off with President Obama over the automatic across-the-board spending cuts that are set to take effect on March 1. And later next month, the stopgap resolution that is keeping the government funded will expire, igniting yet another battle over spending.

Cantor’s new prescription also comes at a time when tensions within the party are rising over how to reorient after losses in November’s election. It is an argument that pits the GOP establishment, which is advocating a more pragmatic course, against insurgent forces such as the tea party, which insist that the best way out of the wilderness is to hew to purist conservative orthodoxy.

In his address to the conservative American Enterprise Institute, Cantor advocated policies aimed at making conservative principles more resonant in the lives of struggling middle-class Americans.

Among his proposals: holding schools more accountable to the concerns of parents, making colleges more attuned to the job market, giving more flexibility to hourly workers who want to spend time with their families, simplifying the tax code and speeding breakthrough medical research.

At a time when many Republicans say the key to future success is winning a greater share of the Hispanic vote, Cantor sounded a more inclusive note on the issue that has alienated many Latinos from the GOP.

“It’s no secret that there are more than 11 million people here illegally, many of whom have become part of the fabric of our country,” he said. “They, like us, have families and dreams.”

Cantor said he would support a path to citizenship for the children of illegal immigrants — the central provision of the Dream Act, which he opposed in the past.

“One of the great founding principles of our country was that children would not be punished for the mistakes of their parents,” he said. “It is time to provide an opportunity for legal residence and citizenship for those who were brought to this country as children and know no other home.”

Cantor’s speech was so different in tone from the truculent rhetoric that normally defines Washington politics that it even drew praise from a prominent Democrat.

“Even though we might have different policy prescriptions than Leader Cantor, Democrats agree with the diagnosis that the shrinking middle class and the accessibility of the American dream are our most pressing challenges,” Sen. Charles E. Schumer (N.Y.) said in a statement issued by his office. “If House Republicans can adapt their agenda to match Leader Cantor’s words, this Congress could surprise people with how productive it can be.”

Fiscal issues have been the GOP’s north star, especially in the years following the presidency of George W. Bush.

In the view of many conservatives, it was a badly needed course correction after the explosion of spending that occurred when Republicans controlled both the White House and Congress.

Outrage over that spending, and the accompanying expansion of government, was one of the factors that spawned the tea party movement and emboldened outside groups such as the Club for Growth to mount aggressive challenges against Republican incumbents.

But the November election suggested that dire predictions of fiscal calamity were not necessarily a message that would appeal to voters across the political spectrum.

Obama was reelected on a specific pledges to raise taxes on the wealthy and to expand federal spending on infrastructure, education and other areas.

“It’s challenging to talk about budgets in a way that resonates with average folks,” said Republican pollster Whit Ayres.

Although most Americans believe that deficits and debt represent a threat to the country’s future, Ayres added, “the challenge is figuring out how to say that in a way that is palatable. Fundamentally, prosperity beats austerity as a message.”

Cantor is not the only Republican suggesting that his party needs to find a more practical way to connect with voters.

In a speech last month at the Republican National Committee’s winter meeting, Louisiana Gov. Bobby Jindal chided his party for having developed what he called an “obsession with zeroes.”

“We seem to have an obsession with government bookkeeping,” he said. “We think if we can just unite behind a proposal to cut the deficit and debt, if we can just put together a spreadsheet and a PowerPoint and a TV ad, all will be well. . . . We as Republicans have to accept that government number-crunching — even conservative number-crunching — is not the answer to our nation’s problems.”