Who Let The Polynomial In

A function is defined as follows: $f(0) = 0$ $f(1) = 1$ $f(n) = f(n - 1) + f(n - 2) + P(n)$ Here, $n\;\epsilon\;N$ and $P(n)$ can be defined as: $P(n) = \sum_{i=0}^{k} \alpha_i n ^ i$, where $\alpha_i\;\epsilon\;I$ is the $ith$ coefficient of the polynomial, counting from $0$. In other words, $P(n)$ is a polynomial of degree $k$ in the variable, $n$. Your task is to compute $f(n)$ for a given