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

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

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

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

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

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

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

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

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

Solving
1 + 3x + 4x2 = 6

Solving for variable 'x'.

Reorder the terms:
1 + -6 + 3x + 4x2 = 6 + -6

Combine like terms: 1 + -6 = -5
-5 + 3x + 4x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-5 + 3x + 4x2 = 0

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

Divide each side by '4'.
-1.25 + 0.75x + x2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 0.75x + 1.25 + x2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 0.75x + x2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 0.75x + x2 = 0 + 1.25
0.75x + x2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
0.75x + x2 = 1.25

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 1.25 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 1.25 + 0.140625

Combine like terms: 1.25 + 0.140625 = 1.390625
0.140625 + 0.75x + x2 = 1.390625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(x + 0.375) equal to 1.179247642 and -1.179247642.

Subproblem 1
x + 0.375 = 1.179247642

Simplifying
x + 0.375 = 1.179247642

Reorder the terms:
0.375 + x = 1.179247642

Solving
0.375 + x = 1.179247642

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = 1.179247642 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 1.179247642 + -0.375
x = 1.179247642 + -0.375

Combine like terms: 1.179247642 + -0.375 = 0.804247642
x = 0.804247642

Simplifying
x = 0.804247642

Subproblem 2
x + 0.375 = -1.179247642

Simplifying
x + 0.375 = -1.179247642

Reorder the terms:
0.375 + x = -1.179247642

Solving
0.375 + x = -1.179247642

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = -1.179247642 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -1.179247642 + -0.375
x = -1.179247642 + -0.375

Combine like terms: -1.179247642 + -0.375 = -1.554247642
x = -1.554247642

Simplifying
x = -1.554247642

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.804247642, -1.554247642}