Polynomial and divisibility

A sequence of polynomials $P_i(x)$ are defined as follows:
$P_1(x) = 1$
$P_2(x) = 1$
$P_{n+2}(x) = (x+2)P_{n+1}(x) - P_n(x), n=1,2,\dots$

Prove that for all $n > 1$, $P_n(x)^2 + x$ is divisible by $P_{n-1}(x)$