Question

use synthetic division to find the quotient.
$\frac{2x^{4}-x^{3}+2x^{2}-4x-2}{x+2}$
$\boldsymbol{?}x^{3}+\boldsymbol{\square}x^{2}+\boldsymbol{\square}x+\boldsymbol{\square}+\frac{\boldsymbol{\square}}{x+2}$

Explanation:

Step1: Identify root & coefficients

Root of divisor $x+2$ is $-2$. Coefficients of dividend: $2, -1, 2, -4, -2$

Step2: Bring down leading coefficient

Bring down $2$ as first quotient term.

Step3: Multiply and add (1st pass)

Multiply $2 \times (-2) = -4$. Add to next coefficient: $-1 + (-4) = -5$

Step4: Multiply and add (2nd pass)

Multiply $-5 \times (-2) = 10$. Add to next coefficient: $2 + 10 = 12$

Step5: Multiply and add (3rd pass)

Multiply $12 \times (-2) = -24$. Add to next coefficient: $-4 + (-24) = -28$

Step6: Multiply and add (4th pass)

Multiply $-28 \times (-2) = 56$. Add to last coefficient: $-2 + 56 = 54$ (remainder)

Answer:

$2x^3 -5x^2 +12x -28 + \frac{54}{x+2}$