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To solve the problem using synthetic division, we are dividing the polynomial \(7 - 6x - 6x^2 - 3x^4\) by \(x + 1\).

### Steps for Synthetic Division:
1. **Write the polynomial in standard form**: Ensure all powers of \(x\) are included, even if their coefficients are zero.
   The standard form of the polynomial is:
   \[
   -3x^4 + 0x^3 - 6x^2 - 6x + 7
   \]

2. **Set up the synthetic division**: 
   Since we are dividing by \(x + 1\), the root is \(x = -1\). We will use this for the synthetic division.

3. **List the coefficients**:
   The coefficients of the polynomial are: 
   \[
   [-3, 0, -6, -6, 7]
   \]

4. **Perform synthetic division**:
   We start by bringing down the first coefficient (\(-3\)) and then multiply it by \(-1\) (the root from \(x + 1\)) and add the result to the next coefficient.

| Coefficients      | -3 | 0  | -6 | -6 | 7  |
|-------------------|----|----|----|----|----|
| Root (\(-1\))     |    |    |    |    |    |
| Synthetic Division |    |    |    |    |    |

- Bring down \(-3\). 
- Multiply \(-3\) by \(-1\), we get \(3\). Add this to the next coefficient \(0\), we get \(3\).
- Multiply \(3\) by \(-1\), we get \(-3\). Add this to the next coefficient \(-6\), we get \(-9\).
- Multiply \(-9\) by \(-1\), we get \(9\). Add this to the next coefficient \(-6\), we get \(3\).
- Multiply \(3\) by \(-1\), we get \(-3\). Add this to the next coefficient \(7\), we get \(4\).

So, the result of the synthetic division is:

\[
[-3, 3, -9, 3, 4]
\]

This corresponds to the polynomial:
\[
-3x^3 + 3x^2 - 9x + 3 + \frac{4}{x+1}
\]

Thus, the quotient is:
\[
-3x^3 + 3x^2 - 9x + 3
\]
with a remainder of \(4\), meaning the final result is:
\[
-3x^3 + 3x^2 - 9x + 3 + \frac{4}{x+1}
\]

Let me know if you want a deeper explanation or have any questions!

Here are 5 related questions for further practice:
1. How does synthetic division differ from long division for polynomials?
2. What would be the quotient if we divide \(4x^3 + 5x^2 - 2x + 1\) by \(x - 2\)?
3. Can synthetic division be applied if the divisor is not linear, e.g., \(x^2 + 1\)?
4. How do you identify the remainder in synthetic division, and what does it represent?
5. What are the steps for performing synthetic division when the divisor is \(x - k\)?

**Tip:** Always remember to arrange the polynomial in descending order of powers of \(x\), and fill in any missing terms with a zero coefficient!

Use synthetic division to divide the polynomials: (7 - 6x - 6x^2 - 3x^4) ÷ (x + 1).

Learn how to divide the polynomial (7 - 6x - 6x^2 - 3x^4) by (x + 1) using synthetic division. This detailed step-by-step solution covers key algebraic concepts and the polynomial remainder theorem, making it ideal for students in Grades 9-12.

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