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To determine which expression is a factor of the polynomial \( x^3 + 2x^2 - 9x - 18 \), we can use the **Rational Root Theorem** or synthetic division.

### Using the Rational Root Theorem:
1. The Rational Root Theorem states that any rational solution (factor) of the polynomial equation \( P(x) = 0 \) is a fraction \( \frac{p}{q} \) where \( p \) is a factor of the constant term (-18) and \( q \) is a factor of the leading coefficient (1).
2. For the polynomial \( x^3 + 2x^2 - 9x - 18 \):
   - Constant term \( p = -18 \) (factors: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \))
   - Leading coefficient \( q = 1 \) (factors: \( \pm 1 \))
3. Possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18 \).

### Testing the possible roots:
- **Test \( x = 2 \):**
  \[
  P(2) = 2^3 + 2 \cdot 2^2 - 9 \cdot 2 - 18
  \]
  \[
  = 8 + 8 - 18 - 18
  \]
  \[
  = 8 + 8 - 36 = 16 - 36 = -20 \quad (\neq 0)
  \]

- **Test \( x = -2 \):**
  \[
  P(-2) = (-2)^3 + 2 \cdot (-2)^2 - 9 \cdot (-2) - 18
  \]
  \[
  = -8 + 8 + 18 - 18
  \]
  \[
  = -8 + 8 + 18 - 18 = 0
  \]
  
Since \( P(-2) = 0 \), \( x = -2 \) is a root of the polynomial, meaning \( (x + 2) \) is a factor.

### Conclusion:
The correct answer is **D. \( (x + 2) \)**.

Would you like a detailed explanation of the steps involved in the Rational Root Theorem or synthetic division?

Here are some related questions you might find helpful:
1. How do you factor a cubic polynomial?
2. What is synthetic division, and how is it used to find polynomial factors?
3. How can you determine all factors of a polynomial?
4. What is the significance of the Rational Root Theorem?
5. How can polynomial division be used to simplify polynomial expressions?

**Tip:** When testing possible roots, start with smaller numbers to save time.

Which expression is a factor of the polynomial x^3 + 2x^2 - 9x - 18?

Explore the process of identifying factors of the polynomial x^3 + 2x^2 - 9x - 18 using the Rational Root Theorem and synthetic division. This article is suitable for high school students studying advanced algebra concepts.

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