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The question asks for an expression of the rational function \( f(x) \) that has a vertical asymptote at \( x = -5 \).

### Key Concept: Vertical Asymptote
A vertical asymptote occurs when the denominator of a rational function equals zero, while the numerator does not. This means that for a vertical asymptote at \( x = -5 \), the denominator must include a factor \( (x + 5) \), which becomes zero when \( x = -5 \).

### Analysis of the Options:
Let’s go through each option to determine if it has \( (x + 5) \) in the denominator, which would produce the required vertical asymptote at \( x = -5 \).

- **Option A:**  
  \( f(x) = \frac{(x - 5)(x + 5)}{2(x - 5)} \)  
  The denominator does not have \( (x + 5) \), so there won’t be an asymptote at \( x = -5 \). This is incorrect.

- **Option B:**  
  \( f(x) = \frac{(x - 4)(x + 5)}{(x - 1)(x + 5)} \)  
  The denominator does have \( (x + 5) \), but it also appears in the numerator. As a result, the \( (x + 5) \) terms cancel out, so there won’t be a vertical asymptote at \( x = -5 \). This is incorrect.

- **Option C:**  
  \( f(x) = \frac{(x + 1)(x + 5)}{(x - 5)(x + 2)} \)  
  The denominator does not include \( (x + 5) \), so there is no vertical asymptote at \( x = -5 \). This is incorrect.

- **Option D (Correct Answer):**  
  \( f(x) = \frac{(x - 5)(x - 3)}{(x - 3)(x + 5)} \)  
  The denominator has \( (x + 5) \), which will lead to a vertical asymptote at \( x = -5 \), as required. Hence, this is the correct choice.

### Conclusion:
The correct answer is **D** because it has the necessary factor \( (x + 5) \) in the denominator, creating the vertical asymptote at \( x = -5 \).

Would you like further details or have any questions? Here are 5 related questions for practice:

1. What does a horizontal asymptote represent in a rational function?
2. How can we determine the horizontal asymptote of a rational function?
3. If the numerator and denominator have a common factor, how does it affect the graph of the rational function?
4. What happens when both the numerator and denominator of a rational function are zero at a certain value of \( x \)?
5. How do we find the x-intercepts of a rational function?

**Tip:** For vertical asymptotes, always focus on the factors of the denominator that do not cancel with the numerator.

In the xy-plane, the graph of a rational function f has a vertical asymptote at x = -5. Which of the following could be an expression for f(x)?

This problem asks to identify which of the given rational function expressions has a vertical asymptote at x = -5. The problem covers core concepts in rational functions, focusing on vertical asymptotes and the behavior of function expressions. Suitable for high school students in Grades 9-12.

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