Find the number of vertical asymptotes in the graph of
\[y = \frac{(x + 8) (x + 5)^2 (x + 1)^3 x^5 (x - 3)^2}{(x + 7) (x + 5)^2 (x + 1) x (x - 3)^3 (x - 4)}.\]
To find the vertical asymptotes of a rational function, I need to look at the factors in the denominator that are not canceled out by factors in the numerator. That means I can ignore the factors of (x + 5)^2, (x + 1), x, and (x - 3)^2, since they appear in both the numerator and the denominator. The remaining factors in the denominator are (x + 7), (x - 3), and (x - 4). These factors will make the denominator zero when x is equal to -7, 3, or 4, respectively. Therefore, the number of vertical asymptotes is the number of these factors, which is 3. 