$f$ entire and $f(z) \in \mathbb{R} \Leftrightarrow z \in \mathbb{R}$ implies $f$ has at most one zero in $\mathbb{C}$

Question

I found this problem on an old Complex Analysis past exam. I have no idea how to approach the problem (don't see how Rouché or anything like that can help me). Any hints are appreciated!