jmc

There are factors of $x+5,$ $x+1,$ and $x$ in both the numerator and denominator, and the factors in the denominator cancel the factors in the numerator, so the graph has a hole at $x=-5,$ $x=-1,$ and $x=0.$

There is a factor of $x+7$ in the denominator, so there is a vertical asymptote at $x=-7.$ There are three factors of $x-3$ in the denominator and two factors of $x-3$ in the numerator, so there is a vertical asymptote at $x=3.$ There is a factor of $x-4$ in the denominator, so there is a vertical asymptote at $x=4.$

Hence, there are $\boxed{3}$ vertical asymptotes.