In the Euclidean Ring ${\Bbb Z}_3[x]$, what is the remainder of the euclidean division of $x^4+2x^3+x^2+2x+2$ by $x^2+2$? So far I've $(x^2+2)(x^2+2x-1)+4-2x$   but the options for the remainder are:
$2x+1$,
$2x$,
$x+1$, or
$x$.

Hint:
$$-2=1\pmod 3\;,\;\;4=1\pmod 3\;$$
