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

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

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

Remove parenthesis around (-2x)
6 + x * -2x + 3x + -4x = 3x + 4

Reorder the terms for easier multiplication:
6 + -2x * x + 3x + -4x = 3x + 4

Multiply x * x
6 + -2x2 + 3x + -4x = 3x + 4

Reorder the terms:
6 + 3x + -4x + -2x2 = 3x + 4

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

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

Solving
6 + -1x + -2x2 = 4 + 3x

Solving for variable 'x'.

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

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

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

Reorder the terms:
2 + -4x + -2x2 = 4 + -4 + 3x + -3x

Combine like terms: 4 + -4 = 0
2 + -4x + -2x2 = 0 + 3x + -3x
2 + -4x + -2x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
2 + -4x + -2x2 = 0

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

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

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

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

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

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

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 2x + x2 = 0 + 1

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

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

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 1 + 1

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

Combine like terms: 1 + 1 = 2
1 + 2x + x2 = 2

Factor a perfect square on the left side:
(x + 1)(x + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
x + 1 = 1.414213562

Simplifying
x + 1 = 1.414213562

Reorder the terms:
1 + x = 1.414213562

Solving
1 + x = 1.414213562

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + x = 1.414213562 + -1
x = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
x = 0.414213562

Simplifying
x = 0.414213562

Subproblem 2
x + 1 = -1.414213562

Simplifying
x + 1 = -1.414213562

Reorder the terms:
1 + x = -1.414213562

Solving
1 + x = -1.414213562

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + x = -1.414213562 + -1
x = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
x = -2.414213562

Simplifying
x = -2.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.414213562, -2.414213562}
Solution
x = {0.414213562, -2.414213562}