Prove that the conjugates of an element over a given extension field are distinct

Let $\theta = i + \sqrt{2})$. Prove that if $e \in \mathbb{Q}$, then the conjugates of $2 - ei + 2\sqrt{2}i$ over $\mathbb{Q}(\theta)$ are distinct.

We know from previous exercises that $\mathbb{Q}(\theta)$ has degree 4 over $\mathbb{Q}$, and that the conjugates of $i + \sqrt{2}$ are $\pm i \pm \sqrt{2}$.

Using some linear algebra, we see that $2 - ei + 2\sqrt{2}i = 1 + \frac{-e}{6}\theta + \theta^2 + \frac{-e}{6}\theta^3$; let $p(x) = 1 - x/6 + x^2 - x^3/6$. Evaluating $p(x)$ at each of the conjugates of $\theta$, we see that the conjugates of $2 - ei + 2\sqrt{2}i$ over $\mathbb{Q}(\theta)$ are $2 - ei + 2\sqrt{2}i$, $2 - ei - 2\sqrt{2}i$, $2 + ei - 2\sqrt{2}i$, and $2 + ei + 2\sqrt{2}i$. Provided $e \neq 0$, these are all distinct.