How to find a factor of $\zeta - 1$ in $\mathbb{Q}(\zeta)$ for $\zeta := e^{2 \pi i / 5}$

Question

Let $\zeta := e^{2 \pi i / 5}$. Given that $\zeta - 1$ is a factor of $\zeta + 4$ in $\mathbb{Z}[\zeta]$, find another factor of $\zeta + 4$.

Remark: This is part of exercise 7 in chapter 3 of Stewart's & Tall's Algebraic Number Theory.

Unfortunately I do not see how to solve this aside frome mere guessing; which was unsuccessfull so far. Could you please give me a hint?