Polynomial Recursion

Number Theory Level 5

A sequence of polynomials \(g_k(x)\) is defined recursively as follows. \[g_0(x)=x; \ g_{k+1}(x)=g_k(x^2+2x)-g_k(x)\] Find the last three digits of the coefficient of \(x^2\) in \(g_{299}(x)\).