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

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

Simplifying
(2x + 1)(1 + -2x)(x + 1) = 1

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

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

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

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

Multiply (1 + -4x2) * (1 + x)
(1(1 + x) + -4x2 * (1 + x)) = 1
((1 * 1 + x * 1) + -4x2 * (1 + x)) = 1
((1 + 1x) + -4x2 * (1 + x)) = 1
(1 + 1x + (1 * -4x2 + x * -4x2)) = 1
(1 + 1x + (-4x2 + -4x3)) = 1
(1 + 1x + -4x2 + -4x3) = 1

Add '-1' to each side of the equation.
1 + 1x + -4x2 + -1 + -4x3 = 1 + -1

Reorder the terms:
1 + -1 + 1x + -4x2 + -4x3 = 1 + -1

Combine like terms: 1 + -1 = 0
0 + 1x + -4x2 + -4x3 = 1 + -1
1x + -4x2 + -4x3 = 1 + -1

Combine like terms: 1 + -1 = 0
1x + -4x2 + -4x3 = 0

Solving
1x + -4x2 + -4x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(1 + -4x + -4x2) = 0

Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(1 + -4x + -4x2)' equal to zero and attempt to solve:

Simplifying
1 + -4x + -4x2 = 0

Solving
1 + -4x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-0.25 + x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x + x2 = 0 + 0.25
x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
x + x2 = 0.25

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

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 0.25 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.25 + 0.25

Combine like terms: 0.25 + 0.25 = 0.5
1.25 + x2 = 0.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.707106781 and -0.707106781.

Subproblem 1
x + 0.5 = 0.707106781

Simplifying
x + 0.5 = 0.707106781

Reorder the terms:
0.5 + x = 0.707106781

Solving
0.5 + x = 0.707106781

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.707106781 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.707106781 + -0.5
x = 0.707106781 + -0.5

Combine like terms: 0.707106781 + -0.5 = 0.207106781
x = 0.207106781

Simplifying
x = 0.207106781

Subproblem 2
x + 0.5 = -0.707106781

Simplifying
x + 0.5 = -0.707106781

Reorder the terms:
0.5 + x = -0.707106781

Solving
0.5 + x = -0.707106781

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.707106781 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.707106781 + -0.5
x = -0.707106781 + -0.5

Combine like terms: -0.707106781 + -0.5 = -1.207106781
x = -1.207106781

Simplifying
x = -1.207106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.207106781, -1.207106781}
Solution
x = {0, 0.207106781, -1.207106781}