Question

1. $(6x^{3}-13x^{2}-12x + 4)div(2x + 1)$

Explanation:

Step1: Divide leading terms

$\frac{6x^3}{2x} = 3x^2$
Multiply divisor by $3x^2$: $3x^2(2x+1)=6x^3+3x^2$
Subtract from dividend:
$(6x^3-13x^2-12x+4)-(6x^3+3x^2) = -16x^2-12x+4$

Step2: Divide new leading terms

$\frac{-16x^2}{2x} = -8x$
Multiply divisor by $-8x$: $-8x(2x+1)=-16x^2-8x$
Subtract from current polynomial:
$(-16x^2-12x+4)-(-16x^2-8x) = -4x+4$

Step3: Divide new leading terms

$\frac{-4x}{2x} = -2$
Multiply divisor by $-2$: $-2(2x+1)=-4x-2$
Subtract from current polynomial:
$(-4x+4)-(-4x-2) = 6$

Answer:

Quotient: $3x^2 - 8x - 2$, Remainder: $6$
Or written as: $3x^2 - 8x - 2 + \frac{6}{2x+1}$