6x-(2x-1)(2x+1)=2-(3+2x^2)

Solution for 6x-(2x-1)(2x+1)=2-(3+2x^2) equation:

Simplifying
6x + -1(2x + -1)(2x + 1) = 2 + -1(3 + 2x2)

Reorder the terms:
6x + -1(-1 + 2x)(2x + 1) = 2 + -1(3 + 2x2)

Reorder the terms:
6x + -1(-1 + 2x)(1 + 2x) = 2 + -1(3 + 2x2)

Multiply (-1 + 2x) * (1 + 2x)
6x + -1(-1(1 + 2x) + 2x * (1 + 2x)) = 2 + -1(3 + 2x2)
6x + -1((1 * -1 + 2x * -1) + 2x * (1 + 2x)) = 2 + -1(3 + 2x2)
6x + -1((-1 + -2x) + 2x * (1 + 2x)) = 2 + -1(3 + 2x2)
6x + -1(-1 + -2x + (1 * 2x + 2x * 2x)) = 2 + -1(3 + 2x2)
6x + -1(-1 + -2x + (2x + 4x2)) = 2 + -1(3 + 2x2)

Combine like terms: -2x + 2x = 0
6x + -1(-1 + 0 + 4x2) = 2 + -1(3 + 2x2)
6x + -1(-1 + 4x2) = 2 + -1(3 + 2x2)
6x + (-1 * -1 + 4x2 * -1) = 2 + -1(3 + 2x2)
6x + (1 + -4x2) = 2 + -1(3 + 2x2)

Reorder the terms:
1 + 6x + -4x2 = 2 + -1(3 + 2x2)
1 + 6x + -4x2 = 2 + (3 * -1 + 2x2 * -1)
1 + 6x + -4x2 = 2 + (-3 + -2x2)

Combine like terms: 2 + -3 = -1
1 + 6x + -4x2 = -1 + -2x2

Solving
1 + 6x + -4x2 = -1 + -2x2

Solving for variable 'x'.

Reorder the terms:
1 + 1 + 6x + -4x2 + 2x2 = -1 + -2x2 + 1 + 2x2

Combine like terms: 1 + 1 = 2
2 + 6x + -4x2 + 2x2 = -1 + -2x2 + 1 + 2x2

Combine like terms: -4x2 + 2x2 = -2x2
2 + 6x + -2x2 = -1 + -2x2 + 1 + 2x2

Reorder the terms:
2 + 6x + -2x2 = -1 + 1 + -2x2 + 2x2

Combine like terms: -1 + 1 = 0
2 + 6x + -2x2 = 0 + -2x2 + 2x2
2 + 6x + -2x2 = -2x2 + 2x2

Combine like terms: -2x2 + 2x2 = 0
2 + 6x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + 3x + -1x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(1 + 3x + -1x2)' equal to zero and attempt to solve:

Simplifying
1 + 3x + -1x2 = 0

Solving
1 + 3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -3x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -3x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3x + x2 = 0 + 1
-3x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3x + x2 = 1

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 1 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + -3x + x2 = 3.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
x + -1.5 = 1.802775638

Simplifying
x + -1.5 = 1.802775638

Reorder the terms:
-1.5 + x = 1.802775638

Solving
-1.5 + x = 1.802775638

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 1.802775638 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.802775638 + 1.5
x = 1.802775638 + 1.5

Combine like terms: 1.802775638 + 1.5 = 3.302775638
x = 3.302775638

Simplifying
x = 3.302775638

Subproblem 2
x + -1.5 = -1.802775638

Simplifying
x + -1.5 = -1.802775638

Reorder the terms:
-1.5 + x = -1.802775638

Solving
-1.5 + x = -1.802775638

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -1.802775638 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.802775638 + 1.5
x = -1.802775638 + 1.5

Combine like terms: -1.802775638 + 1.5 = -0.302775638
x = -0.302775638

Simplifying
x = -0.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.302775638, -0.302775638}
Solution
x = {3.302775638, -0.302775638}