VTRMC 2017 Problem 6

Let $f ( x ) \in \mathbb { Z } [ x ]$ be a polynomial with integer coefficients such that $f ( 1 ) = - 1 , f ( 4 ) = 2$ and $f ( 8 ) = 34$ . Suppose $n\in\mathbb{Z}$ is an integer such that $f ( n ) = n ^ { 2 } - 4 n - 18$ . Determine all possible values for $n$ .