GARDEN CITY, Kan. — State Senator Laura Kelly, the Democrat running for governor of Kansas, likes to talk about Sam Brownback, the Republican ex-governor. At a debate last week, she mentioned him five times. And that was just in her two-minute opening statement.

Democrats see a path to power in red-leaning Kansas this way: Talk a lot about Mr. Brownback and his tax-slashing experiment that left gaping state revenue shortfalls in recent years. Then tie the ex-governor as closely as possible to Secretary of State Kris W. Kobach, the Republican nominee for governor, who has burnished a national profile with ominous warnings about vote fraud and illegal immigration.

Polls have shown an especially close race, and Mr. Brownback, who left office in January to become an ambassador, looms large. His legacy is so problematic that Mr. Kobach has taken pains to distance himself without offending core Republican voters. And his name is so toxic that Ms. Kelly utters it as often as possible: in stump speeches, commercials, campaign emails.

Speaking to the debate crowd at a packed middle school auditorium in Garden City, a Republican stronghold in the state’s rural southwest, Ms. Kelly said, “Everywhere you look: Our schools, our roads, our economy, our health care, all devastated by the Brownback tax experiment.”