Question 6
Let $P(x)$ be a nonconstant polynomial with integer coefficients. Find all polynomials $Q(x)$ with integer coefficients such that, for every positive integer $n$, there is a polynomial $R_n(x)$ with integer coefficients satisfying
$$Q(x)^{2n}-1=R_n(x)\bigl(P(x)^{2n}-1\bigr).$$
$$Q(x)^{2n}-1=R_n(x)\bigl(P(x)^{2n}-1\bigr).$$