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The problem asks to find which expression could represent the rational function \( f(x) \), given that it has zeros at \( x = -2 \) and \( x = -9 \).

### Step 1: Analyze the given zeros
Since the function has zeros at \( x = -2 \) and \( x = -9 \), the numerator of the rational function must be zero at these values. This means the factors of the numerator should be:
\[
(x + 2)(x + 9)
\]
Expanding this:
\[
(x + 2)(x + 9) = x^2 + 11x + 18
\]
Thus, the numerator should be \( x^2 + 11x + 18 \).

### Step 2: Analyze the denominator
We are not given any specific information about the denominator other than it must be one of the options provided. So, we will compare the denominator in each option to find the matching one.

### Step 3: Compare the options
- The numerator should be \( x^2 + 11x + 18 \), which appears in the second and fourth options.
- We will now check the denominators:
  - In the second option: \( x^2 + 7x + 18 \)
  - In the fourth option: \( x^2 - 7x - 18 \)

There is no constraint given for the denominator, so we cannot derive any specific form. However, both the second and fourth options are valid possibilities since the numerator is correct.

Thus, based on the given information, the second option appears to be the correct choice.

Would you like further details or have any questions?

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Here are 5 related questions you might consider:
1. How can we determine the factors of a quadratic expression?
2. What is the importance of the denominator in a rational function?
3. Can a rational function have vertical asymptotes? If so, how do we find them?
4. How do zeros of a rational function affect its graph?
5. How do you simplify rational expressions?

**Tip:** When factoring quadratics, always check if the roots match the zeros given in the problem!

A rational function f(x) has zeros at x = -2 and x = -9. Which of the following expressions could represent f(x)?

Learn how to find a rational function with given zeros at x = -2 and x = -9. This problem covers key algebraic concepts, including factoring quadratics and analyzing rational functions. Suitable for students in Grades 10-12.

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