We've been given a solid region E in $\mathbb{R}^3$, of known volume 10 cubic units.
Now, we're told to calculate the flux of a field $H = \frac{−3}{(x^2+y^2+z^2)^{3/2}} \langle x, y, z \rangle$ through $\partial$E, assuming the origin lies inside E.
The hint given is use Gauss' Law. However, when I proceed to calculate the divergence, it equates to $0$, making the flux $0$. This, unfortunately, is incorrect, as the given answer is $-12\pi$.
Is my understanding of the question wrong?
As pointed out by @DonAntonio, you cannot use the divergence/Gauss theorem here, because the vector field is not differentiable in $(0,0,0)$, which lies in $E$. You can however use a generalization of the theorem, which states that the flux equals $$ \Phi = \iiint_E \nabla \cdot \vec{H}\; dV - \iint_S\vec{H}\cdot d\vec{S} $$ where $S$ is any closed surface in which lies $(0,0,0)$, with an inward orientation. The first term equals $0$, and it is not hard to compute the second one with $S$ being a sphere with a "small" radius. You are going to have to use spherical coordinates, and the field $\vec{H}$ will be more simple to manage. In fact you will notice that the flux through a sphere does not depend on its radius here (the radius will vanish in the computations), leaving you with $$-3\iint_S \sin\phi \; d\theta d\phi = -3 \times 4\pi =-12 \pi$$