-2x(3x+7)=2x+6

Solution for -2x(3x+7)=2x+6 equation:

Simplifying
-2x(3x + 7) = 2x + 6

Reorder the terms:
-2x(7 + 3x) = 2x + 6
(7 * -2x + 3x * -2x) = 2x + 6
(-14x + -6x2) = 2x + 6

Reorder the terms:
-14x + -6x2 = 6 + 2x

Solving
-14x + -6x2 = 6 + 2x

Solving for variable 'x'.

Reorder the terms:
-6 + -14x + -2x + -6x2 = 6 + 2x + -6 + -2x

Combine like terms: -14x + -2x = -16x
-6 + -16x + -6x2 = 6 + 2x + -6 + -2x

Reorder the terms:
-6 + -16x + -6x2 = 6 + -6 + 2x + -2x

Combine like terms: 6 + -6 = 0
-6 + -16x + -6x2 = 0 + 2x + -2x
-6 + -16x + -6x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-6 + -16x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '-2'.
-2(3 + 8x + 3x2) = 0

Ignore the factor -2.
Subproblem 1
Set the factor '(3 + 8x + 3x2)' equal to zero and attempt to solve:

Simplifying
3 + 8x + 3x2 = 0

Solving
3 + 8x + 3x2 = 0

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

Divide each side by '3'.
1 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.666666667x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 2.666666667x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.666666667x + x2 = 0 + -1
2.666666667x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.666666667x + x2 = -1

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = -1 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = -1 + 1.777777780

Combine like terms: -1 + 1.777777780 = 0.77777778
1.777777780 + 2.666666667x + x2 = 0.77777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 0.77777778

Calculate the square root of the right side: 0.881917105

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 0.881917105 and -0.881917105.

Subproblem 1
x + 1.333333334 = 0.881917105

Simplifying
x + 1.333333334 = 0.881917105

Reorder the terms:
1.333333334 + x = 0.881917105

Solving
1.333333334 + x = 0.881917105

Solving for variable 'x'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = 0.881917105 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 0.881917105 + -1.333333334
x = 0.881917105 + -1.333333334

Combine like terms: 0.881917105 + -1.333333334 = -0.451416229
x = -0.451416229

Simplifying
x = -0.451416229

Subproblem 2
x + 1.333333334 = -0.881917105

Simplifying
x + 1.333333334 = -0.881917105

Reorder the terms:
1.333333334 + x = -0.881917105

Solving
1.333333334 + x = -0.881917105

Solving for variable 'x'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = -0.881917105 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -0.881917105 + -1.333333334
x = -0.881917105 + -1.333333334

Combine like terms: -0.881917105 + -1.333333334 = -2.215250439
x = -2.215250439

Simplifying
x = -2.215250439

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.451416229, -2.215250439}
Solution
x = {-0.451416229, -2.215250439}