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?

To add to @Ethan Bolker's answer, I want to address your question about the order of $x$ and $(x+1)$. You are correct about the order of $x$, since 
$$x^8=(x^2)^4=(-2)^4=16=1.$$
However, the order of $x+1$ is not one. This would imply that $(x+1)^2=(x+1)$, which is not true. It should be noted that the order of a generator of a nontrivial group is never 1. The order of $(x+1)$ is 12. You should check this is true. 
