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?

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.