4x(2x+6)=180

Solution for 4x(2x+6)=180 equation:

Simplifying
4x(2x + 6) = 180

Reorder the terms:
4x(6 + 2x) = 180
(6 * 4x + 2x * 4x) = 180
(24x + 8x2) = 180

Solving
24x + 8x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 24x + 8x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 24x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-45 + 6x + 2x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-45 + 6x + 2x2)' equal to zero and attempt to solve:

Simplifying
-45 + 6x + 2x2 = 0

Solving
-45 + 6x + 2x2 = 0

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

Divide each side by '2'.
-22.5 + 3x + x2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 3x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 3x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 3x + x2 = 0 + 22.5
3x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
3x + x2 = 22.5

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 22.5 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 22.5 + 2.25

Combine like terms: 22.5 + 2.25 = 24.75
2.25 + 3x + x2 = 24.75

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 24.75

Calculate the square root of the right side: 4.974937186

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.974937186 and -4.974937186.

Subproblem 1
x + 1.5 = 4.974937186

Simplifying
x + 1.5 = 4.974937186

Reorder the terms:
1.5 + x = 4.974937186

Solving
1.5 + x = 4.974937186

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 4.974937186 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 4.974937186 + -1.5
x = 4.974937186 + -1.5

Combine like terms: 4.974937186 + -1.5 = 3.474937186
x = 3.474937186

Simplifying
x = 3.474937186

Subproblem 2
x + 1.5 = -4.974937186

Simplifying
x + 1.5 = -4.974937186

Reorder the terms:
1.5 + x = -4.974937186

Solving
1.5 + x = -4.974937186

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -4.974937186 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -4.974937186 + -1.5
x = -4.974937186 + -1.5

Combine like terms: -4.974937186 + -1.5 = -6.474937186
x = -6.474937186

Simplifying
x = -6.474937186

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.474937186, -6.474937186}
Solution
x = {3.474937186, -6.474937186}