How do elements really factor in $\mathbb{Z}[\sqrt{-5}]$?

Most undergraduate level abstract algebra texts use $\mathbb{Z}[\sqrt{-5}]$ as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface of how elements actually factor in this ring of algebraic integers. We offer here an interactive framework which shows that while $\mathbb{Z}[\sqrt{-5}]$ is not a UFD, it does satisfy a slightly weaker factorization condition, known as half-factoriality. The arguments involved revolve around the Fundamental Theorem of Ideal Theory in algebraic number fields.