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To solve this problem, we need to determine which function \( g(x) \) has a vertical asymptote at \( x = -5 \). A vertical asymptote in a rational function \( \frac{P(x)}{Q(x)} \) occurs where the denominator \( Q(x) = 0 \), but the numerator \( P(x) \neq 0 \) at that point.

### Step-by-Step Solution

1. **Factor the Denominator of Each Option**: We will focus on the factorization of the denominator in each function. We’re looking for denominators that have a factor of \( x + 5 \), since setting \( x + 5 = 0 \) results in \( x = -5 \), creating a potential asymptote.

2. **Evaluate Each Option**:

   - **Option A: \( g(x) = \frac{x^2 + 9x + 20}{x^2 - x - 12} \)**
     - Denominator: \( x^2 - x - 12 \).
     - Factoring: \( x^2 - x - 12 = (x + 3)(x - 4) \).
     - This does **not** produce \( x + 5 = 0 \), so there is no asymptote at \( x = -5 \).

   - **Option B: \( g(x) = \frac{x^2 + 7x + 12}{x^2 - x - 20} \)**
     - Denominator: \( x^2 - x - 20 \).
     - Factoring: \( x^2 - x - 20 = (x + 5)(x - 4) \).
     - This produces a factor \( x + 5 \), which implies a vertical asymptote at \( x = -5 \).
     - **This option could be correct.**

   - **Option C: \( g(x) = \frac{x^2 - 2x - 15}{x^2 + x - 20} \)**
     - Denominator: \( x^2 + x - 20 \).
     - Factoring: \( x^2 + x - 20 = (x + 5)(x - 4) \).
     - This produces a factor \( x + 5 \), implying a vertical asymptote at \( x = -5 \).
     - **This option could also be correct.**

   - **Option D: \( g(x) = \frac{x^2 + 8x + 15}{x^2 - 7x + 12} \)**
     - Denominator: \( x^2 - 7x + 12 \).
     - Factoring: \( x^2 - 7x + 12 = (x - 3)(x - 4) \).
     - This does **not** produce \( x + 5 = 0 \), so there is no asymptote at \( x = -5 \).

3. **Conclusion**:
   - The correct answers are **Option B** and **Option C**, as both have denominators with a factor of \( x + 5 \), which would produce a vertical asymptote at \( x = -5 \).

Would you like further details on the factorization process or have any questions?

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### Related Questions to Explore

1. How do you determine if a rational function has a hole instead of an asymptote?
2. Why does a factor in the denominator cause a vertical asymptote in rational functions?
3. What are the differences between vertical asymptotes and horizontal asymptotes?
4. How do you determine the horizontal asymptote of a rational function?
5. What conditions create holes in the graph of a rational function?

### Tip
When factoring quadratic expressions, always check for common binomial factors in both the numerator and the denominator to identify potential vertical asymptotes or holes.

In the xy-plane, the graph of a rational function g(x) has a vertical asymptote at x = -5. Which of the following expressions could represent g(x)?

Learn how to identify vertical asymptotes in rational functions by factoring polynomials. This example involves analyzing options to find where the denominator has a factor of x + 5, resulting in an asymptote at x = -5. Suitable for high school math students.

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