How do you find the elements of $\mathbb{F} _5 [x] / (x^2 + 2)$? I am new to the field $\mathbb{F} _5 [x] / (x^2 + 2)$.  How would I find all the elements present in this field?
Additionally, I know that the order of the element $x$ is 8 and the order of element $(1+x)$ is 1 (it is the generator), but how would I prove this?
Edit: the order of $(1+x)$ is the same as the order of the group as it is the generator.  How would I prove this?

$\mathbb{F} _5 [x] / (x^2 + 2) \cong \mathbb{F} _5 [u]$ with $u^2 =-2$.
$\mathbb{F} _5 $ is the set of all polynomials expressions in $u$ with coefficients in $\mathbb{F} _5$. Since $u^2 =-2$, all polynomials expressions can be reduced to the for $a+bu$ with $a,b \in \mathbb{F} _5$.
