To solve this equation, we need to set each factor equal to zero and solve for x.

First factor:
5x - 1 = 0
5x = 1
x = 1/5

Second factor:
4x^2 - 9 = 0
4x^2 = 9
x^2 = 9/4
x = ±√(9/4)
x = ±3/2

Therefore, the solutions to the equation (5x - 1)(4x^2 - 9) = 0 are x = 1/5, x = 3/2, and x = -3/2.