4x(2x+6)=90

Solution for 4x(2x+6)=90 equation:

Simplifying
4x(2x + 6) = 90

Reorder the terms:
4x(6 + 2x) = 90
(6 * 4x + 2x * 4x) = 90
(24x + 8x2) = 90

Solving
24x + 8x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + 24x + 8x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 24x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-45 + 12x + 4x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-45 + 12x + 4x2)' equal to zero and attempt to solve:

Simplifying
-45 + 12x + 4x2 = 0

Solving
-45 + 12x + 4x2 = 0

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

Divide each side by '4'.
-11.25 + 3x + x2 = 0

Move the constant term to the right:

Add '11.25' to each side of the equation.
-11.25 + 3x + 11.25 + x2 = 0 + 11.25

Reorder the terms:
-11.25 + 11.25 + 3x + x2 = 0 + 11.25

Combine like terms: -11.25 + 11.25 = 0.00
0.00 + 3x + x2 = 0 + 11.25
3x + x2 = 0 + 11.25

Combine like terms: 0 + 11.25 = 11.25
3x + x2 = 11.25

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 = 11.25 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 11.25 + 2.25

Combine like terms: 11.25 + 2.25 = 13.5
2.25 + 3x + x2 = 13.5

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 13.5

Calculate the square root of the right side: 3.674234614

Break this problem into two subproblems by setting 
(x + 1.5) equal to 3.674234614 and -3.674234614.

Subproblem 1
x + 1.5 = 3.674234614

Simplifying
x + 1.5 = 3.674234614

Reorder the terms:
1.5 + x = 3.674234614

Solving
1.5 + x = 3.674234614

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 = 3.674234614 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 3.674234614 + -1.5
x = 3.674234614 + -1.5

Combine like terms: 3.674234614 + -1.5 = 2.174234614
x = 2.174234614

Simplifying
x = 2.174234614

Subproblem 2
x + 1.5 = -3.674234614

Simplifying
x + 1.5 = -3.674234614

Reorder the terms:
1.5 + x = -3.674234614

Solving
1.5 + x = -3.674234614

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 = -3.674234614 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -3.674234614 + -1.5
x = -3.674234614 + -1.5

Combine like terms: -3.674234614 + -1.5 = -5.174234614
x = -5.174234614

Simplifying
x = -5.174234614

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.174234614, -5.174234614}
Solution
x = {2.174234614, -5.174234614}