The following holds:

Let $P(x)$ be a polynomial in one variable $x$ of degree $3$ with complex coefficients such that

a) $$ P(-1)=P(1)=0 $$

Then

b) the formal derivative $P^{'}(x)$ has a root in the region

$$ R = \lbrace z : \mid Re(z)\mid < 1 \rbrace $$

of the complex plane.

Want to know if it is possible to prove this known fact by an appropriate application of Rouche's theorem.