On the multiplicative order of the roots of bX^(q^r+1)-aX^(q^r)+dX-c

In this paper, we find a lower bound for the order of the group $\langle \theta+\alpha\rangle \subset \overline{\mathbb F_{q}}^*$, where $\alpha\in \mathbb F_{q}$, $\theta$ is a generic root of the polynomial $F_{A,r}(X)=bX^{q^r+1}-aX^{q^r}+dX-c\in \mathbb F_{q}[X]$ and $ad-bc\ne0$.