Although every statewide elected official in Kansas is a Republican and President Obama lost the state by more than 20 points in the last election, Mr. Brownback’s proudly conservative policies have turned out to be so divisive and his tax cuts have generated such a drop in state revenue that they have caused even many Republicans to revolt. Projections put state budget shortfalls in the hundreds of millions of dollars annually, raising questions of whether the state can adequately fund education in particular.

This has boosted the hopes of the Democratic candidate, Paul Davis, the State House minority leader, who has shot up in the polls even though he has offered few specifics about how he would run the state. Many disaffected Republicans might give Mr. Davis their vote because, if nothing else, he is not Mr. Brownback.

“There’s just a lot of negative momentum behind Brownback, and Davis has been hammering that home,” said Chapman Rackaway, a political-science professor at Fort Hays State University.