The function r(x) = x^2 + 2x - 24/x + 6 is undefined for x = 0 (division by 0 is forbidden), so the hole in the graph corresponds to x = 0. This is also the equation of a vertical asymptote, because r(x) -> +infty as x -> 0- and r(x) -> -infty as x -> 0+.

If the function is written as r(x) = x^2 + 2x - 24 / (x + 6) or r(x) = x^2 + 2(x - 24) / (x + 6), then it has a vertical asymptote x = - 6 and a hole at x = -6.

If it is defined as r(x) = (x^2 + 2x - 24) / (x + 6), then r(x) = (x - 4) (x + 6) / (x + 6) = x - 4 ( x not = -6).

The last function has no vertical asymptotes, its graph is a line y = x - 4 with a hole at x = -6.