Question 6
Let $P(x)$ be a monic polynomial of degree $100$ with $100$ distinct noninteger real roots. Suppose that each of polynomials $P\left(2x^{2}-4x\right)$ and $P\left(4x-2x^{2}\right)$ has exactly $130$ distinct real roots. Prove that there exist non constant polynomials $A(x), B(x)$ such that $A(x)B(x)=P(x)$ and $A(x)=B(x)$ has no root in $(-1 ; 1)$.