1. Stating the problem: Divide the polynomial $2n^3 + 4n^2 - 3n - 6$ by $n + 3$ using synthetic division. 2. Synthetic division requires the divisor in the form $n - c$. Here, $n + 3$ is rewritten as $n - (-3)$, so $c = -3$. 3. Write the coefficients of the dividend polynomial: $2$ (for $n^3$), $4$ (for $n^2$), $-3$ (for $n^1$), and $-6$ (constant term). 4. Set up synthetic division: | -3 | 2 | 4 | -3 | -6 | |----|---|---|----|----| 5. Bring down the first coefficient: 2. 6. Multiply $2$ by $-3$ to get $-6$. Write this under the second coefficient. 7. Add $4 + (-6) = -2$. 8. Multiply $-2$ by $-3$ to get $6$. Write this under the third coefficient. 9. Add $-3 + 6 = 3$. 10. Multiply $3$ by $-3$ to get $-9$. Write this under the fourth coefficient. 11. Add $-6 + (-9) = -15$. 12. The numbers at the bottom row except the last one are the coefficients of the quotient polynomial, starting from degree two since division reduces degree by one. 13. Thus, the quotient is $2n^2 - 2n + 3$ and the remainder is $-15$. 14. Final answer expressed as: $$\frac{2n^3 + 4n^2 - 3n - 6}{n + 3} = 2n^2 - 2n + 3 + \frac{-15}{n + 3}$$