Use Rolle's Theorem and argue the case that has at most one real root in the interval [-1,1]
To use Rolle's theorem to argue there is at most one, you can proceed thusly:
Assume, to the contrary, there are two roots (or more, but at least 2 is fine), say for and , both in the interval you are considering. and you may also assume that . then that means
and so according to Rolle's theorem, there must be a point , such that and .
Where can you get with that?