  Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form
$f(X)=X+aX^{q(q-1)+1}+bX^{2(q-1)+1}\in\Bbb F_{q^2}[X]$, where $q$ is even and
$a,b\in\Bbb F_{q^2}^*$. They found sufficient conditions on $a,b$ for $f$ to be
a permutation polynomial (PP) of $\Bbb F_{q^2}$ and they conjectured that the
sufficient conditions are also necessary. The conjecture has been confirmed by
Bartoli using the Hasse-Weil bound. In this paper, we give an alternative
solution to the question. We also use the Hasse-Weil bound, but in a different
way. Moreover, the necessity and sufficiency of the conditions are proved by
the same approach.
