Polynomial division

Statement

Divide the polynomial $2x^3+3x^2-11x-6$ by the binomial $x-2$, giving the quotient and the remainder.

Solution

Long division (or synthetic division with $x=2$): $2x^3+3x^2-11x-6=(x-2)\,(2x^2+7x+3).$ Checking the coefficients: $2x^2/x$ gives $2x^2$; then $7x$; then $3$; the remainder is $0$.

The quotient factorises further: $2x^2+7x+3=(2x+1)(x+3)$.

$\boxed{\text{quotient } 2x^2+7x+3,\ \text{remainder } 0}$