jmc

An $x$-intercept occurs when $y=0$. So, the $x$-intercepts are the solutions to the equation $0=(x-5)(x^2+5x+6)$. From this equation, we see that solutions occur when $x-5=0$ and when $x^2+5x+6=0$. Now, $x^2+5x+6$ factors to $(x+3)(x+2)$. So, the solutions are $5,-2,-3$, which comes to $\boxed{3}$ intercepts.