(2x-3)(x+5)-2(x-2)=2(x+3)2-1

Solution for (2x-3)(x+5)-2(x-2)=2(x+3)2-1 equation:

Simplifying
(2x + -3)(x + 5) + -2(x + -2) = 2(x + 3) * 2 + -1

Reorder the terms:
(-3 + 2x)(x + 5) + -2(x + -2) = 2(x + 3) * 2 + -1

Reorder the terms:
(-3 + 2x)(5 + x) + -2(x + -2) = 2(x + 3) * 2 + -1

Multiply (-3 + 2x) * (5 + x)
(-3(5 + x) + 2x * (5 + x)) + -2(x + -2) = 2(x + 3) * 2 + -1
((5 * -3 + x * -3) + 2x * (5 + x)) + -2(x + -2) = 2(x + 3) * 2 + -1
((-15 + -3x) + 2x * (5 + x)) + -2(x + -2) = 2(x + 3) * 2 + -1
(-15 + -3x + (5 * 2x + x * 2x)) + -2(x + -2) = 2(x + 3) * 2 + -1
(-15 + -3x + (10x + 2x2)) + -2(x + -2) = 2(x + 3) * 2 + -1

Combine like terms: -3x + 10x = 7x
(-15 + 7x + 2x2) + -2(x + -2) = 2(x + 3) * 2 + -1

Reorder the terms:
-15 + 7x + 2x2 + -2(-2 + x) = 2(x + 3) * 2 + -1
-15 + 7x + 2x2 + (-2 * -2 + x * -2) = 2(x + 3) * 2 + -1
-15 + 7x + 2x2 + (4 + -2x) = 2(x + 3) * 2 + -1

Reorder the terms:
-15 + 4 + 7x + -2x + 2x2 = 2(x + 3) * 2 + -1

Combine like terms: -15 + 4 = -11
-11 + 7x + -2x + 2x2 = 2(x + 3) * 2 + -1

Combine like terms: 7x + -2x = 5x
-11 + 5x + 2x2 = 2(x + 3) * 2 + -1

Reorder the terms:
-11 + 5x + 2x2 = 2(3 + x) * 2 + -1

Reorder the terms for easier multiplication:
-11 + 5x + 2x2 = 2 * 2(3 + x) + -1

Multiply 2 * 2
-11 + 5x + 2x2 = 4(3 + x) + -1
-11 + 5x + 2x2 = (3 * 4 + x * 4) + -1
-11 + 5x + 2x2 = (12 + 4x) + -1

Reorder the terms:
-11 + 5x + 2x2 = 12 + -1 + 4x

Combine like terms: 12 + -1 = 11
-11 + 5x + 2x2 = 11 + 4x

Solving
-11 + 5x + 2x2 = 11 + 4x

Solving for variable 'x'.

Reorder the terms:
-11 + -11 + 5x + -4x + 2x2 = 11 + 4x + -11 + -4x

Combine like terms: -11 + -11 = -22
-22 + 5x + -4x + 2x2 = 11 + 4x + -11 + -4x

Combine like terms: 5x + -4x = 1x
-22 + 1x + 2x2 = 11 + 4x + -11 + -4x

Reorder the terms:
-22 + 1x + 2x2 = 11 + -11 + 4x + -4x

Combine like terms: 11 + -11 = 0
-22 + 1x + 2x2 = 0 + 4x + -4x
-22 + 1x + 2x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-22 + 1x + 2x2 = 0

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

Divide each side by '2'.
-11 + 0.5x + x2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 0.5x + 11 + x2 = 0 + 11

Reorder the terms:
-11 + 11 + 0.5x + x2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 0.5x + x2 = 0 + 11
0.5x + x2 = 0 + 11

Combine like terms: 0 + 11 = 11
0.5x + x2 = 11

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 = 11 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 11 + 0.0625

Combine like terms: 11 + 0.0625 = 11.0625
0.0625 + 0.5x + x2 = 11.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(x + 0.25) equal to 3.326033674 and -3.326033674.

Subproblem 1
x + 0.25 = 3.326033674

Simplifying
x + 0.25 = 3.326033674

Reorder the terms:
0.25 + x = 3.326033674

Solving
0.25 + x = 3.326033674

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 = 3.326033674 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 3.326033674 + -0.25
x = 3.326033674 + -0.25

Combine like terms: 3.326033674 + -0.25 = 3.076033674
x = 3.076033674

Simplifying
x = 3.076033674

Subproblem 2
x + 0.25 = -3.326033674

Simplifying
x + 0.25 = -3.326033674

Reorder the terms:
0.25 + x = -3.326033674

Solving
0.25 + x = -3.326033674

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 = -3.326033674 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -3.326033674 + -0.25
x = -3.326033674 + -0.25

Combine like terms: -3.326033674 + -0.25 = -3.576033674
x = -3.576033674

Simplifying
x = -3.576033674

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.076033674, -3.576033674}