TR-MO 2000 Problem 2

Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$ . Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that $$P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).$$