Quadratic Equation:
To solve the given equation, we have a quadratic equation of the form ax^2 + bx + c = 0, where a = 1, b = -(2b-1), and c = -(b^2-b-20).

Quadratic Formula:
The quadratic formula is used to find the roots of a quadratic equation. It states that the solutions for x can be found using the formula x = (-b ± √(b^2-4ac)) / 2a.

Substituting the values:
Let's substitute the values of a, b, and c into the quadratic formula to find the roots of the equation.

x = [-(2b-1) ± √((2b-1)^2 - 4(1)(-(b^2-b-20))) / (2(1)]

Simplifying the equation:
Let's simplify the equation step by step.

1. Expand (2b-1)^2:
(2b-1)^2 = (2b-1)(2b-1) = 4b^2 - 2b - 2b + 1 = 4b^2 - 4b + 1

2. Expand -4(1)(-(b^2-b-20)):
-4(1)(-(b^2-b-20)) = 4b^2 - 4b + 80

3. Substitute the values back into the quadratic formula:
x = [-(2b-1) ± √(4b^2 - 4b + 1 - 4b^2 + 4b - 80)] / 2

Simplifying further:
Let's simplify the equation inside the square root.

√(4b^2 - 4b + 1 - 4b^2 + 4b - 80) = √(-79)

Since the expression inside the square root is negative, the quadratic equation has no real solutions.

Therefore, the equation X^2 - (2b-1)x - (b^2-b-20) = 0 has no real solutions for x.