A man who could careless...but still looks good, of course!
The word could is a modal verb and a modal verb always modifies a main verb. The word careless is an adjective not a verb. For this reason, I'm sure that the writer of this quiz intended to use the two separate words "care" and "less" with the modal verb could modifying the main verb care.
A man who could care less...but still looks good, of course!
By the way, "could care less" is an idiom that indicates a lack of interest in something. Of course, it should be noted that this idiom originated, and makes much more sense semantically, as "couldn't care less." Think of it this way - if someone "could care less" it means that they do care to at least some extent whereas if someone "couldn't care less" it means that it would be impossible to care less because they do not care at all.
I know that Facebook quizzes do not suite me because they are filled with so many errors that they make my head spin - punctuation, grammar, spelling, capitalization...the list goes on and on.