I want to factorize this polynomial over finite field $F_q$. with $q \equiv 5$ (mod 6), and $q=11$
$$x^4+ x^3 (u+u^{-1}) \rho^{-1} + x^3 (u+u^{-1}) \rho + x^2 \rho^2 + x^2 \rho^{-2} + (u+u^{-1})^2 x^2 + (u+u^{-1}) x \rho^{-1} + (u+u^{-1}) x \rho +1.$$
I know that we can use the formula $ (x^2+ax+b)(x^2+cx+d)$, and then we have
$$a+c = (u+u^{-1}) \rho^{-1} + (u+u^{-1}) \rho $$ $$b+d+ac = \rho^2 + \rho^{-2} + (u+u^{-1})^2$$ $$ad + bc = (u+u^{-1}) \rho^{-1} + (u+u^{-1}) \rho +1$$
then we have $a+c = ad +bc $ I am trying to solve that as $a+c-ad-bc=0$ then If $d=1$, we have $$a+c-a -bc =0$$ $$c-bc=0$$ either $c=0$ or $b=1 $
so I am not sure if I am in the correct way or not. I need to know how to factorize this polynomial to be irreducible over finite field
Thanks..