To simplify the expression, first, find a common denominator for the fractions:

2x + 1 / 2x - 1 - (x + 1) / 2x + 1 = 4 / 4x^2 - 1

Multiply the numerator and denominator of the second fraction by (2x - 1) to get a common denominator:

2x + 1 / 2x - 1 - (x + 1)(2x - 1) / (2x + 1)(2x - 1) = 4 / 4x^2 - 1
2x + 1 / 2x - 1 - (2x^2 - x + 2x - 1) / (4x^2 - 1) = 4 / 4x^2 - 1
2x + 1 / 2x - 1 - (2x^2 + x - 1) / (4x^2 - 1) = 4 / 4x^2 - 1

Combine like terms in the second fraction:

2x + 1 / 2x - 1 - (2x^2 + 2x - 1) / (4x^2 - 1) = 4 / 4x^2 - 1
2x + 1 / 2x - 1 - 2x^2 - 2x + 1 / 4x^2 - 1 = 4 / 4x^2 - 1

Combine like terms in the second fraction:

2x + 1 / 2x - 1 - 2x^2 - 2x + 1 / 4x^2 - 1 = 4 / 4x^2 - 1
2x + 1 / 2x - 1 - 2x^2 - 2x + 1 / 4x^2 - 1 = 4 / (2x + 1)(2x - 1)

Now, we need to have everything on the same denominator:

[2x(2x+1) + 1(2x+1) - (2x^2 + 2x - 1)(2x-1)] / (2x-1)(2x+1) = 4 / (2x+1)(2x-1)

Expanding the numerators, we get:

[4x^2 + 2x + 2x + 1 - (4x^3 - 2x^2 - 4x + 2x^2 - x + 1)] / (2x-1)(2x+1) = 4 / (2x+1)(2x-1)
[6x + 1 - 4x^3 + 4x - 1] / (2x-1)(2x+1) = 4 / (2x+1)(2x-1)
(6x + 1 - 4x^3 + 4x - 1) / (2x-1)(2x+1) = 4 / (2x-1)(2x+1)

Simplify the terms in the numerator:

(10x - 4x^3) / (2x-1)(2x+1) = 4 / (2x-1)(2x+1)

Divide both sides by the denominator:

10x - 4x^3 = 4

Now, we have:

4x^3 - 10x + 4 = 0

This is the final simplified expression.