Find the quotient (6x^2 - 8x - 6) divided by (3x + 2).

Step-by-step solution

1. 1

   Recognise the non-monic divisor. The divisor $3x+2$ leads with $3x$, not $x$, so synthetic division does not apply directly and each quotient term comes from dividing by $3x$ rather than by $x$. Long division handles this without any extra machinery.

2. 2

   First step. $\dfrac{6x^2}{3x}=2x$. Multiply back: $2x(3x+2)=6x^2+4x$. Subtract: $(6x^2-8x)-(6x^2+4x)=-12x$, then bring down $-6$ to get $-12x-6$.

3. 3

   Second step. $\dfrac{-12x}{3x}=-4$. Multiply back: $-4(3x+2)=-12x-8$. Subtract: $(-12x-6)-(-12x-8)=2$.

4. 4

   Stop when the degree drops. The leftover $2$ is a constant, lower in degree than the linear divisor, so the process terminates with quotient $2x-4$ and remainder $2$.

5. 5

   Express the answer. $\dfrac{6x^2-8x-6}{3x+2}=2x-4+\dfrac{2}{3x+2}$.

6. 6

   Check by multiplying back and by substitution. $(3x+2)(2x-4)+2=6x^2-12x+4x-8+2=6x^2-8x-6$. As a second check, the divisor vanishes at $x=-\tfrac23$, and the dividend there is $6\cdot\tfrac49+\tfrac{16}{3}-6=\tfrac83+\tfrac{16}{3}-6=8-6=2$, which is the remainder.