\begin{flushleft}
&=ax(a^2+y^2)+az(a^2+x^2)+ay(a^2+z^2)-az(a^2+y^2)-ax(a^2+z^2)-ay(a^2+x^2)\\
&=a^3x+axy^2+a^3z+azx^2+a^3y+ayz^2-a^3z-azy^2-a^3x-axz^2-a^3y-ayx^2\\
&=a(xy^2+zx^2+yz^2-zy^2-xz^2-x^2y)\\
&=a\Bigl(z(x^2+yz-xz)-y(-x^2+zy-xy)\Bigr)\\
&=a\Bigl(z(x^2+yz)-xz^2-y(x^2+yz)+xy^2\Bigr)\\
&=a\Bigl((x^2+yz)(z-y)-x(z^2-y^2)\Bigr)\\
&=a\Bigl((x^2+yz)(z-y)-x(z-y)(z+y)\Bigr)\\
&=a(z-y)(x^2+yz-xz-xy)\\
&=a(z-y)\Bigl(x(x-y)-z(x-y)\Bigr)\\
&=a(z-y)(x-y)(x-z)
\end{flushleft}