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

Setting 2cosx + 1 = 0:
2cosx = -1
cosx = -1/2

This gives us x = 2π/3 and x = 4π/3 as solutions.

Setting 2sinx - √3 = 0:
2sinx = √3
sinx = √3/2

This gives us x = π/3 and x = 2π/3 as solutions.

Therefore, the solutions to the equation (2cosx + 1)(2sinx - √3) = 0 are x = π/3, 2π/3, and 4π/3.