Determine if the function below has a horizontal or a slant asymptote, and explain why. \[ f(x)=\frac{-13 x-10 x^{2}+5}{-2 x-3} \] Answer Attempt 1 out of 2 The function has a \( \quad \vee \) asymptote because the degree of the numerator is \( \quad \vee \) the degree of the denominator. As \( x \) approaches infinity (positive or negative), the value of \( y \) approaches \( \begin{array}{l}\text {. }\end{array} \)

To determine if the function \( f(x) = \frac{-13x - 10x^2 + 5}{-2x - 3} \) has a horizontal or slant asymptote, we need to compare the degrees of the numerator and the denominator. The degree of the numerator is 2 (highest power of \( x \) is \( x^2 \)), and the degree of the denominator is 1 (highest power of \( x \) is \( x \)). Since the degree of the numerator is greater than the degree of the denominator, the function has a slant asymptote. To find the slant asymptote, we can perform long division of the numerator by the denominator. Let's perform the long division to find the slant asymptote. Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\left(-13x-10x^{2}+5\right)\div \left(-2x-3\right)\) - step1: Calculate: \(\left(5x-1\right)\text{ }\textrm{remainder}\text{ }2\) The long division gives us the quotient \( 5x - 1 \) and a remainder of 2. Therefore, the slant asymptote of the function \( f(x) = \frac{-13x - 10x^2 + 5}{-2x - 3} \) is \( y = 5x - 1 \).