Divide $x^{4}+2x^{3}+3x^{2}+2x+20$ by $x^{2}+2x+2$.

(A) To divide $x^{4}+2x^{3}+3x^{2}+2x+20$ by $x^{2}+2x+2$,we use long division:
$1$. Divide the first term of the dividend $(x^{4})$ by the first term of the divisor $(x^{2})$ to get $x^{2}$.
$2$. Multiply $x^{2}$ by $(x^{2}+2x+2)$ to get $x^{4}+2x^{3}+2x^{2}$.
$3$. Subtract this from the dividend: $(x^{4}+2x^{3}+3x^{2}+2x+20) - (x^{4}+2x^{3}+2x^{2}) = x^{2}+2x+20$.
$4$. Divide the first term of the new polynomial $(x^{2})$ by the first term of the divisor $(x^{2})$ to get $1$.
$5$. Multiply $1$ by $(x^{2}+2x+2)$ to get $x^{2}+2x+2$.
$6$. Subtract this from the current remainder: $(x^{2}+2x+20) - (x^{2}+2x+2) = 18$.
Thus,the quotient is $x^{2}+1$ and the remainder is $18$.