https://file.aiquickdraw.com/mathbot/file/17295512626083oly6zbn.jpg

To solve the given expression using synthetic division, we first need to understand that synthetic division typically works when dividing a polynomial by a binomial of the form \(x - c\). However, the divisor in this case is \(6 + x\), which can be rewritten as \(x + 6\) to fit the typical form.

### Step-by-step synthetic division:
We are tasked with dividing:

\[
(x^4 - 6x^3 - 6x + x^2) \div (x + 6)
\]

Rewriting the polynomial in standard order:

\[
x^4 - 6x^3 + x^2 - 6x + 0
\]

Here are the steps:
1. Write the coefficients of the polynomial: \([1, -6, 1, -6, 0]\)
2. For synthetic division, we use the opposite of the constant in the divisor. Since \(x + 6\) is the divisor, we use \(-6\).
3. Perform the synthetic division process by first bringing down the leading coefficient.

#### Synthetic Division Table:

|  Coefficients  |  1   |  -6   |  1   |  -6  |  0   |
|----------------|------|-------|------|------|------|
| Multiply by -6 |      |  -6   |  72  | -438 | 2592 |
| Add down       |  1   |  -12  |  73  | -438 | 2592 |

### Final Result:
The quotient is:

\[
x^3 - 12x^2 + 73x - 438 + \frac{2592}{x+6}
\]

This matches the result you provided in the image, confirming that the division was done correctly.

### Simplification:
The expression cannot be simplified further, so the final answer is:

\[
x^3 - 12x^2 + 73x - 438 + \frac{2592}{x+6}
\]

Let me know if you'd like more details on any step!

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### Related Questions:
1. How is synthetic division different from long division of polynomials?
2. Can synthetic division be used for all types of polynomial division?
3. What happens if the divisor is a binomial of the form \(ax + b\)?
4. What are some practical uses of synthetic division in real-world applications?
5. How can you verify your answer after performing synthetic division?

### Tip:
When doing synthetic division, always ensure the polynomial is in descending order of degree, and add placeholders for missing terms (like the constant) to avoid errors.

Learn how to divide the polynomial x^4 - 6x^3 + x^2 - 6x by (6 + x) using synthetic division. Step-by-step guide with a detailed synthetic division table and final simplified result. Suitable for high school students.

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