NZ-NZMO 2020 Problem 8

For a positive integer $x$ , define a sequence $a_{0}, a_{1}, a_{2}, \ldots$ according to the following rules: $a_{0} = 1$ , $a_{1} = x + 1$ and

$$a_{n + 2} = x a_{n + 1} - a_{n} \mathrm{for~all~}n\geq 0.$$

Prove that there exist infinitely many positive integers $x$ such that this sequence does not contain a prime number.