(2X-3)(X+1)=-1

Solution for (2X-3)(X+1)=-1 equation:

Simplifying
(2X + -3)(X + 1) = -1

Reorder the terms:
(-3 + 2X)(X + 1) = -1

Reorder the terms:
(-3 + 2X)(1 + X) = -1

Multiply (-3 + 2X) * (1 + X)
(-3(1 + X) + 2X * (1 + X)) = -1
((1 * -3 + X * -3) + 2X * (1 + X)) = -1
((-3 + -3X) + 2X * (1 + X)) = -1
(-3 + -3X + (1 * 2X + X * 2X)) = -1
(-3 + -3X + (2X + 2X2)) = -1

Combine like terms: -3X + 2X = -1X
(-3 + -1X + 2X2) = -1

Solving
-3 + -1X + 2X2 = -1

Solving for variable 'X'.

Reorder the terms:
-3 + 1 + -1X + 2X2 = -1 + 1

Combine like terms: -3 + 1 = -2
-2 + -1X + 2X2 = -1 + 1

Combine like terms: -1 + 1 = 0
-2 + -1X + 2X2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-1 + -0.5X + X2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.5X + 1 + X2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.5X + X2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.5X + X2 = 0 + 1
-0.5X + X2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.5X + X2 = 1

The X term is -0.5X.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5X + 0.0625 + X2 = 1 + 0.0625

Reorder the terms:
0.0625 + -0.5X + X2 = 1 + 0.0625

Combine like terms: 1 + 0.0625 = 1.0625
0.0625 + -0.5X + X2 = 1.0625

Factor a perfect square on the left side:
(X + -0.25)(X + -0.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(X + -0.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
X + -0.25 = 1.030776406

Simplifying
X + -0.25 = 1.030776406

Reorder the terms:
-0.25 + X = 1.030776406

Solving
-0.25 + X = 1.030776406

Solving for variable 'X'.

Move all terms containing X to the left, all other terms to the right.

Add '0.25' to each side of the equation.
-0.25 + 0.25 + X = 1.030776406 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + X = 1.030776406 + 0.25
X = 1.030776406 + 0.25

Combine like terms: 1.030776406 + 0.25 = 1.280776406
X = 1.280776406

Simplifying
X = 1.280776406

Subproblem 2
X + -0.25 = -1.030776406

Simplifying
X + -0.25 = -1.030776406

Reorder the terms:
-0.25 + X = -1.030776406

Solving
-0.25 + X = -1.030776406

Solving for variable 'X'.

Move all terms containing X to the left, all other terms to the right.

Add '0.25' to each side of the equation.
-0.25 + 0.25 + X = -1.030776406 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + X = -1.030776406 + 0.25
X = -1.030776406 + 0.25

Combine like terms: -1.030776406 + 0.25 = -0.780776406
X = -0.780776406

Simplifying
X = -0.780776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {1.280776406, -0.780776406}