  In this article, we establish a sufficient condition for the existence of
primitive element $\alpha\in \Fm$ is such that $f(\alpha)$ is also primitive
element of $\Fm$ and $Tr_{\Fm/\F}(\alpha)=\beta$, for any prescribed
$\beta\in\F$, where $f(x)= ax^2 + bx + c\in \Fm(x)$ such that $b^2-4ac\neq 0$.
We conclude that, for $m\geq 5$ there is only one exceptional pair $(q,m)$
which is $(2,6)$.
