Find the equations of any vertical asymptotes for the function below. \( f(x)=\frac{x^{2}+3 x-10}{x^{2}+2 x-15} \)

To find vertical asymptotes, we need to determine where the denominator is equal to zero, provided the numerator is not also zero at those points. For the function \( f(x) = \frac{x^{2} + 3x - 10}{x^{2} + 2x - 15} \), we start with the denominator: 1. Set the denominator equal to zero: \( x^{2} + 2x - 15 = 0 \). 2. Factor the quadratic: \( (x + 5)(x - 3) = 0 \). 3. Solve for \( x \): The roots are \( x = -5 \) and \( x = 3 \). Next, we need to check if the numerator is non-zero at these points. 1. Evaluate the numerator at \( x = -5 \): \( (-5)^{2} + 3(-5) - 10 = 25 - 15 - 10 = 0 \). Since both numerator and denominator are zero, it's not a vertical asymptote (it may be a hole). 2. Evaluate the numerator at \( x = 3 \): \( (3)^{2} + 3(3) - 10 = 9 + 9 - 10 = 8 \). Since this does not equal zero, \( x = 3 \) is indeed a vertical asymptote. Thus, the function has a vertical asymptote at \( x = 3 \). The point \( x = -5 \) is a hole in the graph.