Solution:
Writing the expression as a single fraction, we have
$$\frac{x^{4} y - x y^{4} + y^{4} z - y z^{4} - z x^{4} + x z^{4}}{(x-y)(x-z)(y-z)}$$
Note that if $x = y$, $x = z$, or $y = z$, then the numerator of the expression above will be $0$. Thus, $(x-y)(x-z)(y-z)$ divides $x^{4} y - x y^{4} + y^{4} z - y z^{4} - z x^{4} + x z^{4}$. Moreover, the numerator can be factored as follows.
$$(x-y)(x-z)(y-z)\left(x^{2} + y^{2} + z^{2} + x y + y z + z x\right)$$
Hence, we are only evaluating $x^{2} + y^{2} + z^{2} + x y + y z + z x$, which is equal to
$$\begin{aligned} \frac{1}{2}\left[(x+y)^{2} + (y+z)^{2} + (z+x)^{2}\right] & = \frac{1}{2}\left[(2 \sqrt{5})^{2} + (2 \sqrt{2})^{2} + (2 \sqrt{3})^{2}\right] \\ & = 2(5 + 2 + 3) \\ & = 20 \end{aligned}$$