Question

(8x³ + 18x² + 19x + 25) ÷ (2x + 3)

Explanation:

Step1: Set up polynomial long division

Divide $8x^3 + 18x^2 + 19x + 25$ by $2x + 3$

Step2: Divide leading terms

$\frac{8x^3}{2x} = 4x^2$. Multiply divisor by $4x^2$:
$4x^2(2x + 3) = 8x^3 + 12x^2$
Subtract from dividend:
$(8x^3 + 18x^2 + 19x + 25) - (8x^3 + 12x^2) = 6x^2 + 19x + 25$

Step3: Divide new leading terms

$\frac{6x^2}{2x} = 3x$. Multiply divisor by $3x$:
$3x(2x + 3) = 6x^2 + 9x$
Subtract from current polynomial:
$(6x^2 + 19x + 25) - (6x^2 + 9x) = 10x + 25$

Step4: Divide new leading terms

$\frac{10x}{2x} = 5$. Multiply divisor by $5$:
$5(2x + 3) = 10x + 15$
Subtract from current polynomial:
$(10x + 25) - (10x + 15) = 10$

Answer:

$4x^2 + 3x + 5 + \frac{10}{2x+3}$