To find the $x$-intercepts of the graph, I need to set $y = 0$ and solve for $x$. So I have $(x-5)(x^2+5x+6) = 0$. This means that either $x-5 = 0$ or $x^2+5x+6 = 0$. If $x-5 = 0$, then $x = 5$, which is one $x$-intercept. If $x^2+5x+6 = 0$, I can factor this quadratic equation as $(x+2)(x+3) = 0$. Then I have $x+2 = 0$ or $x+3 = 0$, which gives me $x = -2$ or $x = -3$, which are two more $x$-intercepts. So in total, I have three distinct $x$-intercepts: $x = -3, -2, 5$.


How many distinct $x$-intercepts does the graph of $y = (x-5)(x^2+5x+6)$ have?