Write an equation for a rational function with the following properties: - Zeros (x-intercepts) at \( x=2 \) and \( x=-6 \). - Vertical asymptotes at \( x=-5 \) and \( x=-2 \). - Horizontal asymptote at \( y=4 \). Leave your equation in factored form. \( r(x)=\square \)

To construct the rational function, proceed as follows: 1. Since the zeros (x-intercepts) are at \( x=2 \) and \( x=-6 \), the numerator must include the factors: \[ (x-2)(x+6) \] 2. The vertical asymptotes occur where the denominator is zero (and these are not canceled by the numerator). Given \( x=-5 \) and \( x=-2 \), the denominator will include: \[ (x+5)(x+2) \] 3. At horizontal asymptotes, we consider the degrees of the numerator and denominator. Here, both have degree 2. The horizontal asymptote is determined by the ratio of the leading coefficients. Without further scaling, the ratio would be \( \frac{1}{1}=1 \), but we want the horizontal asymptote to be \( y=4 \). Therefore, we need to multiply the numerator by 4. Combining these pieces, the rational function in factored form is: \[ r(x)=\frac{4(x-2)(x+6)}{(x+5)(x+2)} \]