(4x+6)(3+5x)=90

Solution for (4x+6)(3+5x)=90 equation:

Simplifying
(4x + 6)(3 + 5x) = 90

Reorder the terms:
(6 + 4x)(3 + 5x) = 90

Multiply (6 + 4x) * (3 + 5x)
(6(3 + 5x) + 4x * (3 + 5x)) = 90
((3 * 6 + 5x * 6) + 4x * (3 + 5x)) = 90
((18 + 30x) + 4x * (3 + 5x)) = 90
(18 + 30x + (3 * 4x + 5x * 4x)) = 90
(18 + 30x + (12x + 20x2)) = 90

Combine like terms: 30x + 12x = 42x
(18 + 42x + 20x2) = 90

Solving
18 + 42x + 20x2 = 90

Solving for variable 'x'.

Reorder the terms:
18 + -90 + 42x + 20x2 = 90 + -90

Combine like terms: 18 + -90 = -72
-72 + 42x + 20x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-72 + 42x + 20x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-36 + 21x + 10x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-36 + 21x + 10x2)' equal to zero and attempt to solve:

Simplifying
-36 + 21x + 10x2 = 0

Solving
-36 + 21x + 10x2 = 0

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

Divide each side by '10'.
-3.6 + 2.1x + x2 = 0.0

Move the constant term to the right:

Add '3.6' to each side of the equation.
-3.6 + 2.1x + 3.6 + x2 = 0.0 + 3.6

Reorder the terms:
-3.6 + 3.6 + 2.1x + x2 = 0.0 + 3.6

Combine like terms: -3.6 + 3.6 = 0.0
0.0 + 2.1x + x2 = 0.0 + 3.6
2.1x + x2 = 0.0 + 3.6

Combine like terms: 0.0 + 3.6 = 3.6
2.1x + x2 = 3.6

The x term is 2.1x.  Take half its coefficient (1.05).
Square it (1.1025) and add it to both sides.

Add '1.1025' to each side of the equation.
2.1x + 1.1025 + x2 = 3.6 + 1.1025

Reorder the terms:
1.1025 + 2.1x + x2 = 3.6 + 1.1025

Combine like terms: 3.6 + 1.1025 = 4.7025
1.1025 + 2.1x + x2 = 4.7025

Factor a perfect square on the left side:
(x + 1.05)(x + 1.05) = 4.7025

Calculate the square root of the right side: 2.168524844

Break this problem into two subproblems by setting 
(x + 1.05) equal to 2.168524844 and -2.168524844.

Subproblem 1
x + 1.05 = 2.168524844

Simplifying
x + 1.05 = 2.168524844

Reorder the terms:
1.05 + x = 2.168524844

Solving
1.05 + x = 2.168524844

Solving for variable 'x'.

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

Add '-1.05' to each side of the equation.
1.05 + -1.05 + x = 2.168524844 + -1.05

Combine like terms: 1.05 + -1.05 = 0.00
0.00 + x = 2.168524844 + -1.05
x = 2.168524844 + -1.05

Combine like terms: 2.168524844 + -1.05 = 1.118524844
x = 1.118524844

Simplifying
x = 1.118524844

Subproblem 2
x + 1.05 = -2.168524844

Simplifying
x + 1.05 = -2.168524844

Reorder the terms:
1.05 + x = -2.168524844

Solving
1.05 + x = -2.168524844

Solving for variable 'x'.

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

Add '-1.05' to each side of the equation.
1.05 + -1.05 + x = -2.168524844 + -1.05

Combine like terms: 1.05 + -1.05 = 0.00
0.00 + x = -2.168524844 + -1.05
x = -2.168524844 + -1.05

Combine like terms: -2.168524844 + -1.05 = -3.218524844
x = -3.218524844

Simplifying
x = -3.218524844

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.118524844, -3.218524844}
Solution
x = {1.118524844, -3.218524844}