Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a\text{ and } b\text{ are both multiple of }5\} \subset \mathbb Z[\sqrt 2]$. I have shown that $I$ is an ideal.
Now I want to show that it is maximal. Since I can write
$$I = 5\mathbb Z[\sqrt 2] = \{ 5(x+y\sqrt 2): x, y, \in \mathbb Z\},$$
I guess $\mathbb Z[\sqrt 2] / I\cong \mathbb Z/5\mathbb Z$. This is a field and thus $I$ is maximal. I am not sure about if the $\cong$ is true though.