$$ \frac{x^3+6x^2-2x-12}{x+6} $$ write than answer in the form of quotient + $$ \frac{\text{remainder}}{x+6} $$

To divide the polynomial $ \frac{x^3 + 6x^2 - 2x - 12}{x + 6} $, we can use polynomial long division.

1. **Divide $ x^3 $ by $ x $** to get $ x^2 $:
   \[
   x^2 \cdot (x + 6) = x^3 + 6x^2
   \]
   Subtract:
   \[
   (x^3 + 6x^2 - 2x - 12) - (x^3 + 6x^2) = -2x - 12
   \]

2. **Divide $ -2x $ by $ x $** to get $ -2 $:
   \[
   -2 \cdot (x + 6) = -2x - 12
   \]
   Subtract:
   \[
   (-2x - 12) - (-2x - 12) = 0
   \]

So the quotient is $ x^2 - 2 $ and the remainder is 0. Therefore, the division yields:

\[
\frac{x^3 + 6x^2 - 2x - 12}{x + 6} = x^2 - 2 + \frac{0}{x + 6}
\]

Thus, the final answer is:

\[
x^2 - 2
\]

Would you like more details or have any questions?

Here are 8 related questions to further your understanding:
1. How do you perform polynomial long division in general?
2. What happens if there is a non-zero remainder in polynomial division?
3. Can synthetic division be used for this problem? If so, how?
4. How do you factorize the numerator polynomial $ x^3 + 6x^2 - 2x - 12 $?
5. What are some applications of polynomial division in real-world problems?
6. How does the degree of the polynomial affect the division process?
7. What are some common mistakes to avoid when performing polynomial division?
8. How can polynomial division be used to simplify rational expressions?

**Tip:** Always ensure that all like terms are correctly aligned when performing polynomial long division to avoid mistakes in subtraction steps.

Learn how to divide the polynomial x^3 + 6x^2 - 2x - 12 by x + 6 using polynomial long division. This problem demonstrates the application of polynomial division in algebra, suitable for Grades 10-12.

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