Necessary and sufficient conditions on the order of a finite field $\mathbb{F}_q$ for the easy identification of primitive polynomials of degree 2

We present the necessary and sufficient conditions on the order $q$ of a finite field $\mathbb{F}_q$ such that every irreducible polynomial of the form $x^2+bx+c \in \mathbb{F}_q[x]$, with $b\neq 0$ and $c$ a primitive element of $\mathbb{F}_q$, is a primitive polynomial.

As a by-product of this result, we also present a new infinite family of finite fields $\mathbb{F}_q$ for which it is easy, in a different way, to determine when an irreducible polynomial of degree two is primitive.