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.

The first assertion can be found without brute force but with a little logic. if $p := x^2+x+4$ is a square then it has a double root (a root with multiplicity 2) so $p$ is divisible by it's derivative $p' = 2x+1$. Euclidean division then gives $x+3$ with, of course, remainder 0.
Let $q_1 \in \Bbb Z_8[x]$ of degree $1$  s.t. $q^2_1 = 1$ then define $q_i = 4x^i + q_{i-1}$. Now $q_i$ has degree $i$ and $q_i^2 = q_{i-1}^2 = 1$. A solution for $q_1$ is $4x+1$. 
