Prime factors of a quadratic expression modulo $4$

Question

I am trying to prove something which needs this to be proved : $4x^2 - 108x + 1$ has not any prime factor of the form $p\equiv 3 (mod 4)$ where $x$ belongs to the sequence $a(n)$ satisfying $a(1) = 34$ , $a(n + 1) = 4a(n)^3 - 104a(n)^2 - 107a(n)$ for all positive integer $n$. I am stuck at this. Any suggestions?