Suppose we have the polynomial $$F_s(x)=cx^{q^s+1}+dx^{q^s}-ax-b \in \mathbb{F}_{q^n}[x]$$ where $ad-bc \neq 0$. The fact that $ad-bc \neq 0$ means that we can take the coefficients of $F_s(x)$ as entries of a matrix $A \in GL(2,q^n)$. This problem was considered by Stichtenoth and Garefalakis for $A=\left( \begin{array}{cc} a&b\\ c&d \end{array}\right) $ and $A=\left( \begin{array}{cc} a&b\\ 0&1 \end{array}\right) $ respectively. In both cases $A$ was taken to be in $GL(2,q)$. In Garefalakis, the exact number of irreducible polynomials of degree say $r$ in the factorization of $F_s(x)$ is obtained.
Now, how I can I find the number of irreducible polynomials(factors) of degree $r$ say in the factorization of $F_s(x)=cx^{q^s+1}+dx^{q^s}-ax-b \in \mathbb{F}_{q^n}[x]$ where the minimal polynomial of $A$ is an irreducible quadratic polynomial over $\mathbb{F}_{q^n}[x]$ where $gcd(n,r)\neq 1$ and $s\mid nr$?