You're a student. One of the other students in your class compliments the professor on his lecture. You think she's just doing that so that she'll get a better grade in the class. You think it's distasteful, so you think this.
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"Suck-up" is an insult for someone who tries too hard to get the favor of an authority figure, like their boss or teacher.
You can call someone "a suck-up":
Henry is such a suck-up. He's always complimenting Kara, like "Oh, what a great idea, Kara. You're so smart." It's disgusting.
Or you can use "suck up" as a verb:
Sometimes you just have to suck up to the boss if you want to get ahead in your career.
This phrase is negative, but it's usually not offensive. If you have friends who are not easily offended, it might be OK to jokingly point out when they are acting like a "suck-up".