grammar - Question mark or period at the end of "What is your favorite song, and explain why"

Which would be correct?

What is your favorite song, and explain why? (I'm thinking this is right.)
What is your favorite
song, and explain why.

This could be written as, "What is your favorite song? Explain why." But that seems kind of clunky. Looking at similar questions on this site I see that maybe there isn't a great answer to this (or maybe that this question is more straight forward than what I was looking at). I dislike the Chicago Manual of Style format, so ideally one of the two choices above would be correct.