How to find the roots of f(x)=x^2+2x+2 in ZZ_3 ? in ZZ_5 ? in R?

How to find the roots of $f\left(x\right)={x}^{2}+2x+2$ in ${\mathbb{Z}}_{3}$ ?
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smh3402en
You can (in each case) start by realizing that
${x}^{2}+2x+2=0$
if and only if
$\left(x+1{\right)}^{2}=-1$
in whatever field you're working in. You've already noted that this doesn't work in
In ${\mathbb{Z}}_{3}$, this becomes
$\left(x+1{\right)}^{2}=2,$
but the only squares in ${\mathbb{Z}}_{3}$ are 0 and 1, so this is impossible. In ${\mathbb{Z}}_{5}$ becomes
$\left(x+1{\right)}^{2}=4.$
Now, ${2}^{2}={3}^{2}=4$, so 1 and 2 are the zeros of f(x) in ${\mathbb{Z}}_{5}$