First, let's distribute and simplify the left side of the equation:

(1-2x)(2x+1) - 4x(2-x)
= 2x^2 + x - 4x + 1 - 8x + 4x
= 2x^2 - 11x + 1

Now, set this expression equal to 0:

2x^2 - 11x + 1 = 0

Next, let's solve this quadratic equation. We can use the quadratic formula:

x = (-(-11) ± sqrt((-11)^2 - 421)) / 2*2
x = (11 ± sqrt(121 - 8)) / 4
x = (11 ± sqrt(113)) / 4

So, the solutions to the equation (1-2x)(2x+1) - 4x(2-x) = 0 are x = (11 + sqrt(113))/4 and x = (11 - sqrt(113))/4.