Cathrene is very careful . . . . . . . . her long hair.

The correct preposition to use in this sentence is about. The sentence should read: "Cathrene is very careful about her long hair." The preposition "about" is used to indicate concern or attention to a particular thing or situation.

The other options are not suitable in this context:
A. "in her long hair" (Incorrect)
B. "at her long hair" (Incorrect)
D. "of her long hair" (Incorrect)