  By adopting the direct method and fixed point method, we prove that the
Hyers-Ulam stability of the following additive-quadratic functional equation
\begin{equation}
  f(x+y, z+w)+f(x-y, z-w)-2 f(x, z)-2 f(x, w)=0 \end{equation} in
$\beta$-homogeneous $F$-spaces and quasi-Banach spaces. There are some
differences that we consider the target space with the $\beta$-homogeneous norm
and quasi-norm. Overcoming the $\beta$-homogeneous norm and quasi-norm
bottlenecks, we get some new results.
