2x(2x+3)=90

Solution for 2x(2x+3)=90 equation:

Simplifying
2x(2x + 3) = 90

Reorder the terms:
2x(3 + 2x) = 90
(3 * 2x + 2x * 2x) = 90
(6x + 4x2) = 90

Solving
6x + 4x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + 6x + 4x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 6x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-45 + 3x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-45 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-45 + 3x + 2x2 = 0

Solving
-45 + 3x + 2x2 = 0

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

Divide each side by '2'.
-22.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + 1.5x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + 1.5x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + 1.5x + x2 = 0 + 22.5
1.5x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
1.5x + x2 = 22.5

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 22.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 22.5 + 0.5625

Combine like terms: 22.5 + 0.5625 = 23.0625
0.5625 + 1.5x + x2 = 23.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 23.0625

Calculate the square root of the right side: 4.802343178

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.802343178 and -4.802343178.

Subproblem 1
x + 0.75 = 4.802343178

Simplifying
x + 0.75 = 4.802343178

Reorder the terms:
0.75 + x = 4.802343178

Solving
0.75 + x = 4.802343178

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 4.802343178 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 4.802343178 + -0.75
x = 4.802343178 + -0.75

Combine like terms: 4.802343178 + -0.75 = 4.052343178
x = 4.052343178

Simplifying
x = 4.052343178

Subproblem 2
x + 0.75 = -4.802343178

Simplifying
x + 0.75 = -4.802343178

Reorder the terms:
0.75 + x = -4.802343178

Solving
0.75 + x = -4.802343178

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -4.802343178 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -4.802343178 + -0.75
x = -4.802343178 + -0.75

Combine like terms: -4.802343178 + -0.75 = -5.552343178
x = -5.552343178

Simplifying
x = -5.552343178

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.052343178, -5.552343178}
Solution
x = {4.052343178, -5.552343178}