Factoring a polynomial over finite field $\,F_3$ that has a root

Question

A question I am struggling with.

We are asked to factor $\,f(x)=x^2+x+1$ over the field $F_3 =\{0,1,2\}$

So, I checked for a root, and I saw that $f(1) = 1^2+1+1 =0$ (because $3=0$ in $F_3$)

that means I can write f(x) as $(x-1)g(x)$ but how do i find $g(x)$?

I did polynomial division and I got that $x^2+x+1$ divided by $x-1$ over $F_3$ is equal to $x+2$

But $f(x)$ is not equal to $(x-1)(x+2)$. I could use some help

Answer 1

$(x-1)(x+2)=x^2+2x-x-2=x^2+x-2=x^2+x+1$ in $F_3$.