2(2x+5)(2x)=106

Solution for 2(2x+5)(2x)=106 equation:

Simplifying
2(2x + 5)(2x) = 106

Reorder the terms:
2(5 + 2x)(2x) = 106

Remove parenthesis around (2x)
2(5 + 2x) * 2x = 106

Reorder the terms for easier multiplication:
2 * 2x(5 + 2x) = 106

Multiply 2 * 2
4x(5 + 2x) = 106
(5 * 4x + 2x * 4x) = 106
(20x + 8x2) = 106

Solving
20x + 8x2 = 106

Solving for variable 'x'.

Reorder the terms:
-106 + 20x + 8x2 = 106 + -106

Combine like terms: 106 + -106 = 0
-106 + 20x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-53 + 10x + 4x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-53 + 10x + 4x2)' equal to zero and attempt to solve:

Simplifying
-53 + 10x + 4x2 = 0

Solving
-53 + 10x + 4x2 = 0

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

Divide each side by '4'.
-13.25 + 2.5x + x2 = 0

Move the constant term to the right:

Add '13.25' to each side of the equation.
-13.25 + 2.5x + 13.25 + x2 = 0 + 13.25

Reorder the terms:
-13.25 + 13.25 + 2.5x + x2 = 0 + 13.25

Combine like terms: -13.25 + 13.25 = 0.00
0.00 + 2.5x + x2 = 0 + 13.25
2.5x + x2 = 0 + 13.25

Combine like terms: 0 + 13.25 = 13.25
2.5x + x2 = 13.25

The x term is 2.5x.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5x + 1.5625 + x2 = 13.25 + 1.5625

Reorder the terms:
1.5625 + 2.5x + x2 = 13.25 + 1.5625

Combine like terms: 13.25 + 1.5625 = 14.8125
1.5625 + 2.5x + x2 = 14.8125

Factor a perfect square on the left side:
(x + 1.25)(x + 1.25) = 14.8125

Calculate the square root of the right side: 3.84870108

Break this problem into two subproblems by setting 
(x + 1.25) equal to 3.84870108 and -3.84870108.

Subproblem 1
x + 1.25 = 3.84870108

Simplifying
x + 1.25 = 3.84870108

Reorder the terms:
1.25 + x = 3.84870108

Solving
1.25 + x = 3.84870108

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = 3.84870108 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = 3.84870108 + -1.25
x = 3.84870108 + -1.25

Combine like terms: 3.84870108 + -1.25 = 2.59870108
x = 2.59870108

Simplifying
x = 2.59870108

Subproblem 2
x + 1.25 = -3.84870108

Simplifying
x + 1.25 = -3.84870108

Reorder the terms:
1.25 + x = -3.84870108

Solving
1.25 + x = -3.84870108

Solving for variable 'x'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + x = -3.84870108 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + x = -3.84870108 + -1.25
x = -3.84870108 + -1.25

Combine like terms: -3.84870108 + -1.25 = -5.09870108
x = -5.09870108

Simplifying
x = -5.09870108

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.59870108, -5.09870108}
Solution
x = {2.59870108, -5.09870108}