  The aim of this paper is to study complete (noncompact) steady
$m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on
the Weyl tensor. In this case, we are able to prove that a steady
$m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional
manifold $(M^{n},g)$, $(n\geq4),$ with nonnegative Ricci curvature and zero
radial Weyl curvature must be a warped product with $(n-1)-$dimensional
Einstein fiber, provided that $M$ has fourth order divergence-free Weyl tensor
(i.e., ${\rm div}^{4}W=0$).
