By the divergence theorem,
[tex]\displaystyle\iint_{\partial\mathcal E}\mathbf f(x,y,z)\cdot\mathrm d\mathbf S=\iiint_{\mathcal E}\nabla\cdot\mathbf f(x,y,z)\,\mathrm dx\,\mathrm dy\,\mathrm dz[/tex]
where [tex]\partial\mathcal E[/tex] is the boundary of [tex]\mathcal E[/tex]. We have
[tex]\nabla\cdot\mathbf f(x,y,z)=\dfrac{\partial(4x)}{\partial x}+\dfrac{\partial(xy)}{\partial y}+\dfrac{\partial(4xz)}{\partial z}=4+x+4x=5x+4[/tex]
so the flux is
[tex]\displaystyle\int_{z=0}^{z=2}\int_{y=0}^{y=2}\int_{x=0}^{x=2}(5x+4)\,\mathrm dx\,\mathrm dy\,\mathrm dz=72[/tex]
