(2x-3)(4x+5)>0

For the product of two numbers to be positive (that is, > 0), either they are both positive or both negative.

Case 1:  both positive:

     2x - 3 > 0     and     4x + 5 > 0

           2x > 3                    4x > -5

            x > 3/2    and         x > -5/4      --->   x > 3/2

Case 2:  both negative:

     2x - 3  <  0     and     4x + 5  <  0

           2x <  3                      4x  <  -5

             x  <  3     and          x <  -5/4     --->  x <  -5/4

Answer:  all the values of x that are either less than -5/4 or greater than 3/2.