Divide $x^{4}+4 x^{3}-2 x^{2}-12 x+9$ by $x^{2}-2 x+1$.

(N/A) To divide $x^{4}+4 x^{3}-2 x^{2}-12 x+9$ by $x^{2}-2 x+1$,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 $(x^2-2x+1)$ to get $x^4-2x^3+x^2$. Subtract this from the dividend to get $6x^3-3x^2-12x+9$.
$3$. Divide $6x^3$ by $x^2$ to get $6x$. Multiply $6x$ by $(x^2-2x+1)$ to get $6x^3-12x^2+6x$. Subtract this to get $9x^2-18x+9$.
$4$. Divide $9x^2$ by $x^2$ to get $9$. Multiply $9$ by $(x^2-2x+1)$ to get $9x^2-18x+9$. Subtract this to get a remainder of $0$.
Thus,the quotient is $x^{2}+6 x+9$ and the remainder is $0$.