Question

divide by using synthetic division. simplify your answer as much as possible.
$\frac{8a^{3}+18a^{2}-a-15}{2a+3} = \square$

Explanation:

Step1: Find root of divisor

Set $2a+3=0$, solve for $a$:
$a = -\frac{3}{2}$

Step2: Set up synthetic division

Use coefficients of dividend $8a^3+18a^2-a-15$: $8, 18, -1, -15$, and root $-\frac{3}{2}$:

$$\begin{array}{r|rrrr} -\frac{3}{2} & 8 & 18 & -1 & -15 \\ & & -12 & -9 & 15 \\ \hline & 8 & 6 & -10 & 0 \end{array}$$

Step3: Form quotient polynomial

The resulting coefficients $8, 6, -10$ correspond to $8a^2+6a-10$. Divide by leading coefficient of divisor $2$:
$\frac{8a^2+6a-10}{2} = 4a^2+3a-5$

Answer:

$4a^2+3a-5$