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

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

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

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

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

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

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

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

Combine like terms: -2x + 3x = 1x
8(-6 + 1x + x2) = 6
(-6 * 8 + 1x * 8 + x2 * 8) = 6
(-48 + 8x + 8x2) = 6

Solving
-48 + 8x + 8x2 = 6

Solving for variable 'x'.

Reorder the terms:
-48 + -6 + 8x + 8x2 = 6 + -6

Combine like terms: -48 + -6 = -54
-54 + 8x + 8x2 = 6 + -6

Combine like terms: 6 + -6 = 0
-54 + 8x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-27 + 4x + 4x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-27 + 4x + 4x2)' equal to zero and attempt to solve:

Simplifying
-27 + 4x + 4x2 = 0

Solving
-27 + 4x + 4x2 = 0

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

Divide each side by '4'.
-6.75 + x + x2 = 0

Move the constant term to the right:

Add '6.75' to each side of the equation.
-6.75 + x + 6.75 + x2 = 0 + 6.75

Reorder the terms:
-6.75 + 6.75 + x + x2 = 0 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x + x2 = 0 + 6.75
x + x2 = 0 + 6.75

Combine like terms: 0 + 6.75 = 6.75
x + x2 = 6.75

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 = 6.75 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 6.75 + 0.25

Combine like terms: 6.75 + 0.25 = 7
1.25 + x2 = 7

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.645751311 and -2.645751311.

Subproblem 1
x + 0.5 = 2.645751311

Simplifying
x + 0.5 = 2.645751311

Reorder the terms:
0.5 + x = 2.645751311

Solving
0.5 + x = 2.645751311

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 = 2.645751311 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.645751311 + -0.5
x = 2.645751311 + -0.5

Combine like terms: 2.645751311 + -0.5 = 2.145751311
x = 2.145751311

Simplifying
x = 2.145751311

Subproblem 2
x + 0.5 = -2.645751311

Simplifying
x + 0.5 = -2.645751311

Reorder the terms:
0.5 + x = -2.645751311

Solving
0.5 + x = -2.645751311

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 = -2.645751311 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.645751311 + -0.5
x = -2.645751311 + -0.5

Combine like terms: -2.645751311 + -0.5 = -3.145751311
x = -3.145751311

Simplifying
x = -3.145751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.145751311, -3.145751311}
Solution
x = {2.145751311, -3.145751311}