2x(2x+2)=6

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

Simplifying
2x(2x + 2) = 6

Reorder the terms:
2x(2 + 2x) = 6
(2 * 2x + 2x * 2x) = 6
(4x + 4x2) = 6

Solving
4x + 4x2 = 6

Solving for variable 'x'.

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

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

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

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

Simplifying
-3 + 2x + 2x2 = 0

Solving
-3 + 2x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x + x2 = 0 + 1.5
x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
x + x2 = 1.5

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 = 1.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
1.25 + x2 = 1.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
x + 0.5 = 1.322875656

Simplifying
x + 0.5 = 1.322875656

Reorder the terms:
0.5 + x = 1.322875656

Solving
0.5 + x = 1.322875656

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 = 1.322875656 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.322875656 + -0.5
x = 1.322875656 + -0.5

Combine like terms: 1.322875656 + -0.5 = 0.822875656
x = 0.822875656

Simplifying
x = 0.822875656

Subproblem 2
x + 0.5 = -1.322875656

Simplifying
x + 0.5 = -1.322875656

Reorder the terms:
0.5 + x = -1.322875656

Solving
0.5 + x = -1.322875656

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 = -1.322875656 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.322875656 + -0.5
x = -1.322875656 + -0.5

Combine like terms: -1.322875656 + -0.5 = -1.822875656
x = -1.822875656

Simplifying
x = -1.822875656

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.822875656, -1.822875656}
Solution
x = {0.822875656, -1.822875656}