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To solve this problem using synthetic division, we are working with the following setup:

1. The divisor is given by \((x - (-4))\) or \(x + 4\).
2. The coefficients of the polynomial are listed at the top row: \(3, 5, -2, -7, 9\).

Let's go through the steps of synthetic division to determine the quotient and remainder.

### Steps of Synthetic Division:

1. **Bring down** the first coefficient, \(3\), to the bottom row.

2. **Multiply** this value \(3\) by the divisor root \(-4\), giving \(-12\). Place this result below the next coefficient.

3. **Add** the second coefficient \(5\) and \(-12\), yielding \(-7\). Write \(-7\) in the bottom row.

4. Repeat the **multiply and add** process:
   - Multiply \(-7\) by \(-4\), resulting in \(28\).
   - Add \(-2\) and \(28\), giving \(26\).
   - Multiply \(26\) by \(-4\), resulting in \(-104\).
   - Add \(-7\) and \(-104\), giving \(-111\).
   - Multiply \(-111\) by \(-4\), resulting in \(444\).
   - Add \(9\) and \(444\), giving \(453\).

So, the final row of synthetic division results is:
\[ 3, -7, 26, -111, 453 \]

### Result
The quotient \(q(x)\) is \(3x^3 - 7x^2 + 26x - 111\), and the remainder \(p(a)\) is \(453\).

The result in the form \(p(x) = (x + 4)q(x) + p(a)\) is:

\[
p(x) = (x + 4)(3x^3 - 7x^2 + 26x - 111) + 453
\]

Would you like further details on each step? 

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Here are five related questions:

1. What is synthetic division and how is it used for polynomial division?
2. How does synthetic division differ from traditional long division of polynomials?
3. How can the remainder theorem be used to find the remainder quickly?
4. What are some practical applications of synthetic division in mathematics?
5. How does the root of the divisor impact the synthetic division process?

**Tip:** In synthetic division, always ensure the polynomial's terms are in descending order and include all powers of \(x\), even if their coefficients are zero.

Use synthetic division to determine the quotient and the remainder. Find the result in the form p(x) = (x - a)q(x) + p(a).

Learn how to use synthetic division to determine the quotient and remainder for a polynomial expression. This solution demonstrates the process of dividing polynomials and provides a step-by-step explanation for finding the result in the form p(x) = (x - a)q(x) + p(a). Suitable for high school students in Grades 10-12.

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