Find the horizontal asymptote of \( f(x)=6 \frac{(x+4)(5 x-1)}{(4-x)(2 x+2)} \)

To find the horizontal asymptote of the function \( f(x)=6 \frac{(x+4)(5 x-1)}{(4-x)(2 x+2)} \), we start by examining the degrees of the numerator and the denominator. The function can be rewritten to analyze the degrees: - The numerator is \( (x+4)(5x-1) \), which is a quadratic polynomial (degree 2). - The denominator is \( (4-x)(2x+2) \), which is also a quadratic polynomial (degree 2). Since both the numerator and the denominator are of the same degree, we can find the horizontal asymptote by taking the leading coefficients of both polynomials. 1. The leading term of \( (x+4)(5x-1) \) is \( 5x^2 \). 2. The leading term of \( (4-x)(2x+2) \) is \( -2x^2 \) (because the \( -x \) results in a negative leading coefficient). Now, the horizontal asymptote is given by the ratio of these leading coefficients: \[ y = \frac{5}{-2} = -\frac{5}{2} \] Thus, the horizontal asymptote of the function \( f(x) \) is \[ y = -\frac{5}{2}. \]