Find all square roots of $x^2+x+4$ in $\mathbb{Z}_5[x]$. 
Find all square roots of $x^2+x+4$ in $\mathbb{Z}_5[x]$. Also show that in $\mathbb{Z}_8[x]$ there are infinitely many square roots of $1$. 

I know how to find the square roots when in $\mathbb{Z}[x]$ however I keep getting confused with $\mathbb{Z}_5[x]$. From here I am assuming I will be able to figure out exactly why the second statement is true.

Hints:
A square root of $x^2+x+4$ has the form $\pm x+a$. Let's identify:
$$(\pm x+a)^2=x^2\pm 2a x+a^2=x^2+x+4.$$
We have to solve $\;\begin{cases}a^2=4\\\pm2a=1\end{cases}$.
The first equation has solutions $\;a=\pm 2$. Checking the second equation, we find the square roots
$$x-2\enspace\text{or}\enspace -x+2. $$
For the second question, try  binomials $ax^n+b$, where the leading coefficient $a$ is nilpotent.
