= Solution
The <ellipsoid> and the exponential factor are invariant under separate reflections of any coordinate. Split the integrand into its two polynomial terms. The term $x^4y^2z$ times that exponential is odd in $z$, so its <volume integral> vanishes by reflecting $z\mapsto-z$. The other term $xy^2z^4$ times the exponential is odd in $x$, so its <volume integral> vanishes by reflecting $x\mapsto-x$. Both integrands are continuous on a bounded region. Therefore <odd-integrand cancellation by reflection> gives
$$
\boxed{\iiint_V(x^4y^2z+xy^2z^4)e^{-(x/a)^2-(y/b)^2-(z/c)^2}\,dx\,dy\,dz=0.}
$$
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