WASHINGTON (AP) — In case the public weren't frustrated enough over Congress' failure to resolve the "fiscal cliff," consider this: lawmakers probably could enact a compromise quickly and easily if Republican leaders let Democrats provide most of the votes.

That would give Democrats a bigger voice in the bargain, of course, which the Republican-led House is loath to do. That's why about 10 percent of the House's members — staunch anti-tax conservatives — were able to thwart Speaker John Boehner's bid to pass a narrowly crafted bill that might have strengthened his bargaining hand.

By trying to pass his plan with GOP votes alone, Boehner could afford to lose only two dozen of the 241 House Republicans. His private headcount found nearly twice that many defectors, party insiders say, forcing Boehner to give up without seeking a formal vote. The miscalculation left negotiations in disarray as the Dec. 31 deadline nears.

The House's 192 Democrats essentially sat on the sidelines, bit players in last week's House drama.

House speakers traditionally advance major legislation only if most of their party's members support it. It's called the "majority of the majority" rule of thumb. But past speakers, including Democrat Nancy Pelosi and Republican Dennis Hastert, have ignored the rule at times.

Some Democrats are now calling on Boehner to do the same to avert the "fiscal cliff" of big tax hikes and spending cuts scheduled to take effect in the new year.

Veterans in both parties say many House Democrats would likely join a sufficient number of Republicans — though not necessarily a majority — to pass a compromise along the lines that Boehner and President Barack Obama seemed to be nearing last week before Boehner struck out on his own. Such a compromise plan might preserve Bush-era tax cuts for all couples making less than $400,000 or so a year. It also could include an outline for future spending cuts and changes to keep the Alternative Minimum Tax from hitting millions of new taxpayers.

While Obama did not explicitly embrace such a plan, he and Boehner appeared to be edging toward some variation of it. But Boehner abruptly launched his separate proposal — he dubbed it "Plan B" — which the conservatives' revolt killed late Thursday.

Boehner's plan would have spared anyone making less than $1 million a year from a tax rate hike in 2013. At least 40 House Republicans refused to back any tax rate increase at all, lawmakers said, dooming the plan.

However, perhaps as many as 200 House Republicans apparently were willing to let tax cuts expire for a fraction of the wealthiest households. That's a significant break from the party's no-tax-hikes-ever orthodoxy.

GOP leaders note that virtually everyone's taxes will rise Jan. 1 without congressional action. They say sparing 98 or 99 percent of Americans is the best political alternative, given Obama's negotiating strength.

It's not known how many House Republicans would vote to avert the "fiscal cliff" by supporting a tax-and-spending plan closer to Obama's liking. In his re-election campaign, the president called for letting the Bush-era tax cuts expire on incomes above $250,000 for couples and $200,000 for individuals. In later negotiations with Boehner, Obama said he could accept a $400,000 threshold if Republicans agreed to other tax and spending provisions.

An Obama-backed deal presumably would draw the overwhelming support of House and Senate Democrats. But even if every House Democrat backed it, it would need another 25 votes in that chamber. Some Democrats say Boehner should let a few dozen willing Republicans provide those votes by putting an Obama-blessed bill on the floor.

"If Speaker Boehner is willing to bring to the floor of the House a bill, and just let this House work its will, Democrats and Republicans voting as their conscience determines, then I believe we can get something done," Rep. Chris Van Hollen, D-Md., told Bloomberg TV.

He noted that Pelosi, as House speaker in 2006, violated the "majority of the majority" rule by letting Republicans provide most of the votes for an Iraq war funding measure she disliked.

Hastert, the Republican speaker from 1999 to 2007, overrode the rule at least twice. In one case, he let Democratic votes carry the load on a stem cell research bill everyone knew President George W. Bush would veto. Hastert also yielded to pressure to let the McCain-Feingold campaign finance bill pass even though most House Republicans opposed it.

Aides to Boehner say he believes in, and abides by, the "majority of the majority" rule without declaring it an iron-clad requirement. They did not respond to messages asking if Boehner might consider ignoring the rule to avert the "fiscal cliff."

John Feehery, who was a top aide to Hastert, said he doubted that would happen.

Boehner "could put something on the floor and vote against it," Feehery said. But it's more likely, he said, that Boehner and other Republicans will let the Dec. 31 deadline pass. That would send the government over the cliff and let virtually all tax rates rise, at least for a while.

Then early next year, Feehery said, lawmakers could "vote to cut taxes rather than vote to raise taxes."

Either way, the richest Americans will see their tax cuts expire. Boehner's hold on his speakership, however, might be stronger if he makes no break in the "majority of the majority" rule.