flati

One root, flat-, and nothing to take apart. The fl- opening tempts nobody into a decode, but the word is worth a page for the meaning rather than the morphology, because English speakers habitually use "flatter" for two different things and Esperanto is stricter.

Flati is praise offered because it will get you something. It is transitive and the object is the person: li flatas la estron, he flatters the boss. The noun is flato and the flatterer is a flatulo or, more usually, flatanto. When the praise is honest you want laŭdi instead, to praise, which the course has not reached yet but which you will meet constantly; when it is a specific compliment you want komplimenti. Keep them apart and your Esperanto gets noticeably less ambiguous than your English.

The second English sense, the mirror kind (this photograph flatters me, it makes me look better than I am), works in Esperanto too and lands the same way: la portreto flatas lin. That is the sense the passive participle carries in the third example above, and the sting of the sentence is that being flatata is something a fair number of people are actively shopping for.

A small usage note that saves an error later: flati takes the person, not the thing said. You flatter somebody, you do not flatter a compliment. If you want the words to be the subject, they take the person as object too, exactly as in the second example.