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To solve this using synthetic division, we'll divide the polynomial \( x^4 + 15x^3 - 2x^2 + 7x + 24 \) by \( x + 3 \).

### Step 1: Set up the synthetic division.
First, rewrite the divisor \( x + 3 \) as \( x - (-3) \). In synthetic division, we'll use \( -3 \) as the divisor.

The coefficients of the polynomial \( x^4 + 15x^3 - 2x^2 + 7x + 24 \) are: 
\[ 1, 15, -2, 7, 24 \]

### Step 2: Perform the synthetic division.

The process of synthetic division involves the following steps:

1. Write down the coefficients of the polynomial in a row: 
   \[ 1 \quad 15 \quad -2 \quad 7 \quad 24 \]
   
2. Bring the first coefficient down as it is.

3. Multiply that number by \( -3 \) (the divisor) and write the result under the next coefficient.

4. Add the second coefficient and the result from step 3.

5. Repeat this process for all the coefficients.

#### Synthetic Division Table:
|        | 1   | 15  | -2  | 7   | 24  |
|--------|-----|-----|-----|-----|-----|
| Multiply by -3: |     | -3  | -36 | 114 | -333 |
| Add:   | 1   | 12  | -38 | 121 | -309 |

### Step 3: Write the quotient and remainder.
The last row (after adding) gives us the coefficients of the quotient and the remainder. The quotient is formed from the numbers \( 1, 12, -38, 121 \), and the remainder is \( -309 \).

Thus, the quotient is:
\[
q(x) = x^3 + 12x^2 - 38x + 121
\]

And the remainder is:
\[
r(x) = -309
\]

### Final answer in the required form:
\[
\frac{x^4 + 15x^3 - 2x^2 + 7x + 24}{x + 3} = (x^3 + 12x^2 - 38x + 121) + \frac{-309}{x+3}
\]

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Would you like more details on any step or further clarifications?

Here are five questions to extend your understanding:
1. What is the difference between synthetic division and long division?
2. How do you know when the remainder is zero in synthetic division?
3. Can synthetic division be used for divisors other than linear polynomials?
4. How would you factor the quotient obtained in this problem?
5. Why is synthetic division more efficient than long division for certain polynomials?

**Tip:** Always check if the divisor is a linear binomial in the form \( x - c \) before using synthetic division!

Use synthetic division to find the result when x^4 + 15x^3 - 2x^2 + 7x + 24 is divided by x + 3.

Solve the polynomial division of x^4 + 15x^3 - 2x^2 + 7x + 24 by x + 3 using synthetic division. This step-by-step solution involves key algebraic concepts such as synthetic division and the polynomial remainder theorem. Suitable for students in Grades 9-12.

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