The ring is $\mathbb{Z_3}[x]$. I am trying to find the $\gcd (x^2-x+1, x^3+2x+2)$. So, $a_1 = x^3+2x+2 \in \mathbb{Z_3}[x]$ and $a_2 = x^2-x+1 \in \mathbb{Z_3}[x]$. By Euclidean algorithm for $a_1 = a_2 \times q_1 + a_3$, I am getting $q_1=x+1$ and $a_3=2x$. Then for $a_2 = a_3 \times q_2 + a_4$, I am getting $q_2=2x+1$ and $a_4=1$ (I used $x=4x$ in $\mathbb{Z_3}[x]$). Then for $a_3 = a_4 \times q_3 + a_5$, I am getting $q_3=2x$ and $a_5=0$. So $a_4=1 = \gcd (x^2-x+1, x^3+2x+2)$. On the other hand, $a_2=x^2-x+1 =(2x+2)^2$ and $a_1=x(2x+2)^2+2x+2$ so $\gcd (x^2-x+1, x^3+2x+2)$ must be $2x+2$ which is not equal to $1$ in $\mathbb{Z_3}[x]$. (!)
How the contradiction happens?
$a_1$ from the book is: $a_1=x(2x+2)^2+2x+2=x^3-x^2+x+2x+2=x^3-x^2+2=x^3+2x^2+2$
While your $a_1$ is $x^3+2x+2$
So they are different polynomials.