(Adds Cruz wins Idaho)
By John Whitesides
DETROIT, March 8 (Reuters) - Republican front-runner Donald Trump rolled to primary wins in the big prize of Michigan and in Mississippi on Tuesday, brushing off a week of blistering attacks from the party's establishment and expanding his lead in the White House nominating race.
Trump's convincing win in Michigan restored his outsider campaign's momentum and increased the pressure on the party's anti-Trump forces to find a way to stop his march to the nomination ahead of several key contests next week.
In the Democratic race, Bernie Sanders stunned front-runner Hillary Clinton in a narrow Michigan primary upset, giving his upstart campaign new energy. Clinton won in Mississippi, but the Sanders win is likely to ensure a prolonged nominating fight.
Trump built his victories in Michigan, in the heart of the industrial Midwest, and Mississippi in the Deep South with broad appeal across many demographics. He won evangelical Christians, Republicans, independents, those who wanted an outsider and those who said they were angry about how the federal government is working, according to exit polls.
At a news conference afterward, Trump said he was drawing new voters to the Republican Party and the establishment figures who are resisting his campaign should save their money and focus on beating the Democrats in November.
"I hope Republicans will embrace it," Trump said of his campaign. "We have something going that is so good, we should grab each other and unify the party."
The results were a setback for rival John Kasich, governor of Ohio, who hoped to pull off a surprise win in neighboring Michigan, and Marco Rubio, a U.S. senator from Florida who has become the establishment favorite but lagged badly in both Michigan and Mississippi and appeared unlikely to win delegates in either.
Trump said Rubio's recent attacks on him had backfired.
"Hostility works for some people; it doesn't work for everyone," Trump said at a news conference in Jupiter, Florida.
Ted Cruz, a U.S. senator from Texas whose recent victories have positioned him as the prime alternative to the brash billionaire, won the party's primary in Idaho.
But Trump suggested his rivals had little hope going forward and took particular aim at Cruz, 45.
"Ted is going to have a hard time," Trump said of Cruz. "He rarely beats me."
The Michigan victory sets Trump up for a potentially decisive day of voting a week from Tuesday. On March 15, Ohio, Florida, Illinois, Missouri and North Carolina - like Michigan, states rich in the delegates who will select their party's nominee at July's Republican National Convention - cast ballots.
The Republican contests in Florida and Ohio award all the state's delegates to the winner. If Trump, 69, could sweep those two states and pile up delegates elsewhere next week, it could knock home-state favorites Rubio and Kasich out of the race and make it tough for Cruz to catch him.
"The biggest takeaway is that the Republican establishment is in its death throes. The only remaining candidates are 100 percent anti-establishment," said Mark Meckler, an early Tea Party movement founder.
Republicans were also voting on Tuesday in caucuses in Hawaii.
Many mainstream Republicans have been offended by Trump's statements on Muslims, immigrants and women and alarmed by his threats to international trade deals. Trump said on Tuesday he has not assembled a foreign policy team, despite having said he would have one in place by February, and dismissed criticism his statements would be harmful to U.S. interests.
Anti-Trump Super PACS have spent millions of dollars on advertisements designed to attack Trump's character in Florida, a state Rubio calls home and Trump calls a second home.
But Trump's relentless anti-free trade rhetoric and promise to slap taxes on cars and parts shipped in from Mexico resonated in Michigan, which has lost tens of thousands of manufacturing and auto industry jobs.
Michigan was the state that spawned the term "Reagan Democrats" to refer to largely white, working-class voters who abandoned their party to vote Ronald Reagan into the White House in the 1980s.
Sal Isabella, a Dearborn insurance agent, said he was for Trump because he would make things happen.
"He'll be like Reagan," Isabella said. "He'll make some big changes and we need big changes.
On the Democratic side, Sanders told reporters in Florida that the results in Michigan had been a repudiation of the opinion polls and the pundits who had written off his chances in the state. Polls had shown Clinton with a double-digit lead going into the primary.
The U.S. senator from Vermont said it showed his political revolution was "strong in every part of the country. Frankly, we believe our strongest areas are yet to come."
Clinton's campaign signaled ahead of Michigan that the race could be tight. Clinton, her husband, former President Bill Clinton, and daughter Chelsea Clinton all campaigned in the state over the past few days trying to garner last-minute votes.
Additional reporting by Emily Stephenson, Alana Wise and Amanda Becker in Washington, and Ginger Gibson in Concord, North Carolina; Editing by Peter Cooney, Frances Kerry and Jonathan Oatis