(x+5)(x-9)(x+3) > 0

The critical numbers are numbers which when substituted for x 
causes the left side either equal to 0 or undefined.

These critical numbers are -5, 9, and -3

Mark those with open circles on the number line:

----------o-----o-----------------------------------o------------
-8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 13

We choose a number less than -5, say -6, and substitute it in the 
inequality:

   (x+5)(x-9)(x+3) > 0
(-6+5)(-6-9)(-6+3) > 0
     (-1)(-15)(-3) > 0
               -45 > 0

That's false, so we do not shade the number line left of -5 

We choose a number between -5 and -3, say -4, and substitute it in the 
inequality:

   (x+5)(x-9)(x+3) > 0
(-4+5)(-4-9)(-4+3) > 0
      (1)(-13)(-1) > 0
                13 > 0

That's true, so we do shade the number line between -5 and -3:

----------o=====o-----------------------------------o------------
-8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 13 

We choose a number between -3 and 9, say 0, and substitute it in the 
inequality:

   (x+5)(x-9)(x+3) > 0
   (0+5)(0-9)(0+3) > 0
        (5)(-9)(3) > 0
              -135 > 0

That's false, so we do not shade the number line between -3 and 9:

We choose a number greater than 9, say 10, and substitute it in the 
inequality:

   (x+5)(x-9)(x+3) > 0
(10+5)(10-9)(10+3) > 0
       (15)(1)(13) > 0
               195 > 0

That's true, so we do shade the number line right of 9 
 
----------o=====o-----------------------------------o============>
-8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 13

Answer:   {x| -5 < x < -3 OR x > 9 }

Edwin