90(x+24)+90(x)=4x(x+4)

Solution for 90(x+24)+90(x)=4x(x+4) equation:

Simplifying
90(x + 24) + 90(x) = 4x(x + 4)

Reorder the terms:
90(24 + x) + 90(x) = 4x(x + 4)
(24 * 90 + x * 90) + 90(x) = 4x(x + 4)
(2160 + 90x) + 90(x) = 4x(x + 4)

Combine like terms: 90x + 90x = 180x
2160 + 180x = 4x(x + 4)

Reorder the terms:
2160 + 180x = 4x(4 + x)
2160 + 180x = (4 * 4x + x * 4x)
2160 + 180x = (16x + 4x2)

Solving
2160 + 180x = 16x + 4x2

Solving for variable 'x'.

Combine like terms: 180x + -16x = 164x
2160 + 164x + -4x2 = 16x + 4x2 + -16x + -4x2

Reorder the terms:
2160 + 164x + -4x2 = 16x + -16x + 4x2 + -4x2

Combine like terms: 16x + -16x = 0
2160 + 164x + -4x2 = 0 + 4x2 + -4x2
2160 + 164x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
2160 + 164x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(540 + 41x + -1x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(540 + 41x + -1x2)' equal to zero and attempt to solve:

Simplifying
540 + 41x + -1x2 = 0

Solving
540 + 41x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-540 + -41x + x2 = 0

Move the constant term to the right:

Add '540' to each side of the equation.
-540 + -41x + 540 + x2 = 0 + 540

Reorder the terms:
-540 + 540 + -41x + x2 = 0 + 540

Combine like terms: -540 + 540 = 0
0 + -41x + x2 = 0 + 540
-41x + x2 = 0 + 540

Combine like terms: 0 + 540 = 540
-41x + x2 = 540

The x term is -41x.  Take half its coefficient (-20.5).
Square it (420.25) and add it to both sides.

Add '420.25' to each side of the equation.
-41x + 420.25 + x2 = 540 + 420.25

Reorder the terms:
420.25 + -41x + x2 = 540 + 420.25

Combine like terms: 540 + 420.25 = 960.25
420.25 + -41x + x2 = 960.25

Factor a perfect square on the left side:
(x + -20.5)(x + -20.5) = 960.25

Calculate the square root of the right side: 30.987900865

Break this problem into two subproblems by setting 
(x + -20.5) equal to 30.987900865 and -30.987900865.

Subproblem 1
x + -20.5 = 30.987900865

Simplifying
x + -20.5 = 30.987900865

Reorder the terms:
-20.5 + x = 30.987900865

Solving
-20.5 + x = 30.987900865

Solving for variable 'x'.

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

Add '20.5' to each side of the equation.
-20.5 + 20.5 + x = 30.987900865 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + x = 30.987900865 + 20.5
x = 30.987900865 + 20.5

Combine like terms: 30.987900865 + 20.5 = 51.487900865
x = 51.487900865

Simplifying
x = 51.487900865

Subproblem 2
x + -20.5 = -30.987900865

Simplifying
x + -20.5 = -30.987900865

Reorder the terms:
-20.5 + x = -30.987900865

Solving
-20.5 + x = -30.987900865

Solving for variable 'x'.

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

Add '20.5' to each side of the equation.
-20.5 + 20.5 + x = -30.987900865 + 20.5

Combine like terms: -20.5 + 20.5 = 0.0
0.0 + x = -30.987900865 + 20.5
x = -30.987900865 + 20.5

Combine like terms: -30.987900865 + 20.5 = -10.487900865
x = -10.487900865

Simplifying
x = -10.487900865

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {51.487900865, -10.487900865}
Solution
x = {51.487900865, -10.487900865}