Chapter 8.2, Problem 12E

For $f\left(x\right)$, $g\left(x\right)$, and ${\text{Z}}_n\left[x\right]$ given in Exercises 11-14, find $s\left(x\right)$ and $t\left(x\right)$ in ${\text{Z}}_n\left[x\right]$ such that $d\left(x\right)=f\left(x\right)s\left(x\right)+g\left(x\right)t\left(x\right)$ is the greatest common divisor of $f\left(x\right)$ and $g\left(x\right)$.

$f\left(x\right)=2x^3+x^2+1$, $g\left(x\right)=x^5+x^4+2x^2+1$, in ${\text{Z}}_3\left[x\right]$