  Let $\mathbb{F}_{q^n}$ be the extension of the field $\mathbb{F}_q$ of degree
n, where $q$ is power of prime $p$, i.e $q=p^k$, where k is a positive integer.
In this paper, we provide sufficient condition for the existence of a primitive
normal element $\alpha\in\mathbb{F}_{q^n} $ such that $\alpha^2+\alpha+1$ is
also primitive normal element over $\mathbb{F}_{q^n}$.
