2x(3x+5)=90

Solution for 2x(3x+5)=90 equation:

Simplifying
2x(3x + 5) = 90

Reorder the terms:
2x(5 + 3x) = 90
(5 * 2x + 3x * 2x) = 90
(10x + 6x2) = 90

Solving
10x + 6x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + 10x + 6x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 10x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-45 + 5x + 3x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-45 + 5x + 3x2)' equal to zero and attempt to solve:

Simplifying
-45 + 5x + 3x2 = 0

Solving
-45 + 5x + 3x2 = 0

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

Divide each side by '3'.
-15 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 1.666666667x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 1.666666667x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 1.666666667x + x2 = 0 + 15
1.666666667x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
1.666666667x + x2 = 15

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 15 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 15 + 0.6944444447

Combine like terms: 15 + 0.6944444447 = 15.6944444447
0.6944444447 + 1.666666667x + x2 = 15.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 15.6944444447

Calculate the square root of the right side: 3.961621441

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 3.961621441 and -3.961621441.

Subproblem 1
x + 0.8333333335 = 3.961621441

Simplifying
x + 0.8333333335 = 3.961621441

Reorder the terms:
0.8333333335 + x = 3.961621441

Solving
0.8333333335 + x = 3.961621441

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 3.961621441 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 3.961621441 + -0.8333333335
x = 3.961621441 + -0.8333333335

Combine like terms: 3.961621441 + -0.8333333335 = 3.1282881075
x = 3.1282881075

Simplifying
x = 3.1282881075

Subproblem 2
x + 0.8333333335 = -3.961621441

Simplifying
x + 0.8333333335 = -3.961621441

Reorder the terms:
0.8333333335 + x = -3.961621441

Solving
0.8333333335 + x = -3.961621441

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -3.961621441 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -3.961621441 + -0.8333333335
x = -3.961621441 + -0.8333333335

Combine like terms: -3.961621441 + -0.8333333335 = -4.7949547745
x = -4.7949547745

Simplifying
x = -4.7949547745

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.1282881075, -4.7949547745}
Solution
x = {3.1282881075, -4.7949547745}