(2x+1)(4x-3)=(6x+1)

Solution for (2x+1)(4x-3)=(6x+1) equation:

Simplifying
(2x + 1)(4x + -3) = (6x + 1)

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

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

Multiply (1 + 2x) * (-3 + 4x)
(1(-3 + 4x) + 2x * (-3 + 4x)) = (6x + 1)
((-3 * 1 + 4x * 1) + 2x * (-3 + 4x)) = (6x + 1)
((-3 + 4x) + 2x * (-3 + 4x)) = (6x + 1)
(-3 + 4x + (-3 * 2x + 4x * 2x)) = (6x + 1)
(-3 + 4x + (-6x + 8x2)) = (6x + 1)

Combine like terms: 4x + -6x = -2x
(-3 + -2x + 8x2) = (6x + 1)

Reorder the terms:
-3 + -2x + 8x2 = (1 + 6x)

Remove parenthesis around (1 + 6x)
-3 + -2x + 8x2 = 1 + 6x

Solving
-3 + -2x + 8x2 = 1 + 6x

Solving for variable 'x'.

Reorder the terms:
-3 + -1 + -2x + -6x + 8x2 = 1 + 6x + -1 + -6x

Combine like terms: -3 + -1 = -4
-4 + -2x + -6x + 8x2 = 1 + 6x + -1 + -6x

Combine like terms: -2x + -6x = -8x
-4 + -8x + 8x2 = 1 + 6x + -1 + -6x

Reorder the terms:
-4 + -8x + 8x2 = 1 + -1 + 6x + -6x

Combine like terms: 1 + -1 = 0
-4 + -8x + 8x2 = 0 + 6x + -6x
-4 + -8x + 8x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
-4 + -8x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-1 + -2x + 2x2) = 0

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

Simplifying
-1 + -2x + 2x2 = 0

Solving
-1 + -2x + 2x2 = 0

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

Divide each side by '2'.
-0.5 + -1x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1x + x2 = 0 + 0.5
-1x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1x + x2 = 0.5

The x term is -1x.  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.
-1x + 0.25 + x2 = 0.5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
0.25 + -1x + x2 = 0.75

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + -0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
x + -0.5 = 0.866025404

Simplifying
x + -0.5 = 0.866025404

Reorder the terms:
-0.5 + x = 0.866025404

Solving
-0.5 + x = 0.866025404

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 = 0.866025404 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 0.866025404 + 0.5
x = 0.866025404 + 0.5

Combine like terms: 0.866025404 + 0.5 = 1.366025404
x = 1.366025404

Simplifying
x = 1.366025404

Subproblem 2
x + -0.5 = -0.866025404

Simplifying
x + -0.5 = -0.866025404

Reorder the terms:
-0.5 + x = -0.866025404

Solving
-0.5 + x = -0.866025404

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 = -0.866025404 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -0.866025404 + 0.5
x = -0.866025404 + 0.5

Combine like terms: -0.866025404 + 0.5 = -0.366025404
x = -0.366025404

Simplifying
x = -0.366025404

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.366025404, -0.366025404}
Solution
x = {1.366025404, -0.366025404}