Question

divide.
$\frac{x^{4}+14}{x+2}$
$x^{3}+?x^{2}+x+ + \frac{}{x+2}$

Explanation:

Step1: Divide leading terms

$\frac{x^4}{x} = x^3$

Step2: Multiply divisor by $x^3$

$x^3(x+2) = x^4 + 2x^3$

Step3: Subtract from dividend

$(x^4 + 14) - (x^4 + 2x^3) = -2x^3 + 14$

Step4: Divide new leading terms

$\frac{-2x^3}{x} = -2x^2$

Step5: Multiply divisor by $-2x^2$

$-2x^2(x+2) = -2x^3 -4x^2$

Step6: Subtract from current remainder

$(-2x^3 +14) - (-2x^3 -4x^2) = 4x^2 +14$

Step7: Divide new leading terms

$\frac{4x^2}{x} = 4x$

Step8: Multiply divisor by $4x$

$4x(x+2) = 4x^2 +8x$

Step9: Subtract from current remainder

$(4x^2 +14) - (4x^2 +8x) = -8x +14$

Step10: Divide new leading terms

$\frac{-8x}{x} = -8$

Step11: Multiply divisor by $-8$

$-8(x+2) = -8x -16$

Step12: Subtract to get final remainder

$(-8x +14) - (-8x -16) = 30$

Answer:

$x^3 + \boldsymbol{-2}x^2 + \boldsymbol{4}x + \boldsymbol{-8} + \frac{\boldsymbol{30}}{x+2}$