To solve the inequality (x-4)(x+5) < 0, we can use the sign chart method.

First, let's find the critical points by setting each factor equal to zero:

x - 4 = 0
x = 4

x + 5 = 0
x = -5

Now, we can create a sign chart using these critical points:

-∞ -5 4 +∞
---------------------
(x-4) - - + +
(x+5) - + + +

Next, we multiply the signs of each factor together to determine the sign of the product:

-∞ -5 4 +∞
---------------------
(x-4) - - + +
(x+5) - + + +
+ - + -

From the sign chart, we can see that the product (x-4)(x+5) is negative when x is between -5 and 4. Therefore, the solution to the inequality is -5 < x < 4.