Chapter 8.2, Problem 4E

For $f\left(x\right)$, $g\left(x\right)$, and ${\text{Z}}_n\left[x\right]$ given in Exercises 1-6, find $q\left(x\right)$ and $r\left(x\right)$ in ${\text{Z}}_n\left[x\right]$ that satisfy the conditions in the Division Algorithm.

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