Find the Remainder (x^4+2x^3-13x^2-12x+21)/(x^2-4x+3)

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of ${\displaystyle 0}$.

Divide the highest order term in the dividend ${\displaystyle x^4}$ by the highest order term in divisor ${\displaystyle x^2}$.

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in ${\displaystyle x^4-4x^3+3x^2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Pull the next terms from the original dividend down into the current dividend.

Divide the highest order term in the dividend ${\displaystyle 6x^3}$ by the highest order term in divisor ${\displaystyle x^2}$.

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in ${\displaystyle 6x^3-24x^2+18x}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Pull the next terms from the original dividend down into the current dividend.

Divide the highest order term in the dividend ${\displaystyle 8x^2}$ by the highest order term in divisor ${\displaystyle x^2}$.

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in ${\displaystyle 8x^2-32x+24}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

The final answer is the quotient plus the remainder over the divisor.