I've come across this here: for a proof that x+1 is a generator.
\begin{align*} &x+1, \quad (x+1)^2=x^2+2x+1=2x, \\ &(x+1)^3=2x(x+1)=2x^2+2x=2x-2=2x+1\\ &(x+1)^4=(2x+1)(x+1)=2x^2+3x+1=2. \end{align*}
But I'm not sure I understand why in \begin{align*} &(x+1)^3=2x(x+1)=2x^2+2x=2x-2=2x+1\\ \end{align*} \begin{align*} 2x^2+2x=2x-2\\ \end{align*}
I tried dividing, subtracting, I multiplied all factors and made a table for Z3[x] x^2.