Answer
$$f\left( x \right) = \frac{{{x^2}}}{{x + 5}}$$
Work Step by Step
$$\eqalign{
& {\text{Vertical asymptote: }}x = - 5 \cr
& {\text{To obtain a vertical asymptote at }}x = - 5,{\text{ we must set in the}} \cr
& {\text{denominator of the rational function an expression whose}} \cr
& {\text{real root is }}x = - 5,{\text{ Using }}x + 5 \cr
& f\left( x \right) = \frac{2}{{x + 5}} \cr
& {\text{Horizontal asymptote: none}} \cr
& {\text{The degree of the numerator must be greater than }} \cr
& {\text{the denominator }}x + 5{\text{ }}\left( {{\text{linear function}}} \right),{\text{ so the denominator}} \cr
& {\text{should be a quadratic function of a higher degree}}{\text{. Let the }} \cr
& {\text{numerator be equal to }}{x^2} \cr
& f\left( x \right) = \frac{{{x^2}}}{{x + 5}} \cr} $$
