To solve this equation, we can begin by expanding both sides of the equation:

Expanding the left side:
(2x - 1)^2 * (x + 5)
= (2x - 1)(2x - 1)(x + 5)
= (4x^2 - 2x - 2x + 1)(x + 5)
= (4x^2 - 4x + 1)(x + 5)
= 4x^3 + 20x^2 - 4x^2 - 20x + x + 5
= 4x^3 + 16x^2 - 19x + 5

Expanding the right side:
(4x + 5)(x + 1)^2
= (4x + 5)(x^2 + 2x + 1)
= 4x^3 + 8x^2 + 4x + 5x^2 + 10x + 5
= 4x^3 + 13x^2 + 14x + 5

Therefore, the equation becomes:
4x^3 + 16x^2 - 19x + 5 = 4x^3 + 13x^2 + 14x + 5

Subtracting 4x^3 from both sides:
16x^2 - 19x + 5 = 13x^2 + 14x + 5

Subtracting 13x^2 and 14x from both sides:
3x^2 - 33x = 0

Factor out an x:
x(3x - 33) = 0

Setting each factor to zero gives the solutions:
x = 0
3x - 33 = 0
3x = 33
x = 11

Therefore, the solutions to the equation are x = 0 and x = 11.