To solve this equation, we will first simplify the expression on the left side of the equation:

2(3x + 1/2)^3 - 9/4 = (-5/2)^4 / (-5/2)^3

First, simplify the left side of the equation:

2(3x + 1/2)^3 - 9/4 = 2(27x^3 + 27x^2/2 + 9x/4 + 1/8) - 9/4
= 54x^3 + 27x^2 + 9x/2 + 1/4 - 9/4
= 54x^3 + 27x^2 + 9x/2 - 2

Now, simplify the right side of the equation:

(-5/2)^4 / (-5/2)^3 = (-5/2)^(4-3)
= (-5/2)^1
= -5/2

Now, set the left side of the equation equal to the right side and solve for x:

54x^3 + 27x^2 + 9x/2 - 2 = -5/2

Multiplying both sides by 2 to get rid of the fraction:

108x^3 + 54x^2 + 9x - 4 = -5

Rearranging the terms:

108x^3 + 54x^2 + 9x + 1 = 0

This is a cubic equation that can be solved using various methods such as factoring, synthetic division, or using the cubic formula.