I'm not in a math class (haven't been in years) but if this question about $\textbf{Z}[\sqrt{-5}]$ appears in some textbook, I wouldn't be surprised.
What I have done: $$1 \times 7 + (3 - \sqrt{-5})(3 + \sqrt{-5}) = 21$$ shows that $\langle 3 + \sqrt{-5} \rangle \subset \langle 7, 4 + \sqrt{-5} \rangle$ and $$(-1) \times 7 + (4 - \sqrt{-5})(4 + \sqrt{-5}) = 14$$ shows that $\langle 4 + \sqrt{-5} \rangle \subset \langle 7, 3 + \sqrt{-5} \rangle$.
But I have this nagging feeling I have ignored some small but important detail.
Signs, plus or minus. That is the small but important detail you have overlooked. Try $$1 \times 7 + (-1)(4 - \sqrt{-5}) = 3 + \sqrt{-5}.$$ Thus $3 + \sqrt{-5} \in \langle 7, 4 - \sqrt{-5} \rangle$; $\langle 7, 3 + \sqrt{-5} \rangle = \langle 7, 4 - \sqrt{-5} \rangle$ and $\langle 7, 3 - \sqrt{-5} \rangle = \langle 7, 4 + \sqrt{-5} \rangle$.
$7$ the number is irreducible though not prime in $\mathbb{Z}[\sqrt{-5}]$. But $\langle 7 \rangle$ the ideal has unique factorization into prime ideals, though those prime ideals may be represented in infinitely many different ways.
The demon's in the details. Mwahahahahahahahaha!