8(x-5)(x+20)+4(x+12)(x-4)=32(x-6)(x+30)

Solution for 8(x-5)(x+20)+4(x+12)(x-4)=32(x-6)(x+30) equation:

Simplifying
8(x + -5)(x + 20) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)

Reorder the terms:
8(-5 + x)(x + 20) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)

Reorder the terms:
8(-5 + x)(20 + x) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)

Multiply (-5 + x) * (20 + x)
8(-5(20 + x) + x(20 + x)) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)
8((20 * -5 + x * -5) + x(20 + x)) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)
8((-100 + -5x) + x(20 + x)) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)
8(-100 + -5x + (20 * x + x * x)) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)
8(-100 + -5x + (20x + x2)) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)

Combine like terms: -5x + 20x = 15x
8(-100 + 15x + x2) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)
(-100 * 8 + 15x * 8 + x2 * 8) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)
(-800 + 120x + 8x2) + 4(x + 12)(x + -4) = 32(x + -6)(x + 30)

Reorder the terms:
-800 + 120x + 8x2 + 4(12 + x)(x + -4) = 32(x + -6)(x + 30)

Reorder the terms:
-800 + 120x + 8x2 + 4(12 + x)(-4 + x) = 32(x + -6)(x + 30)

Multiply (12 + x) * (-4 + x)
-800 + 120x + 8x2 + 4(12(-4 + x) + x(-4 + x)) = 32(x + -6)(x + 30)
-800 + 120x + 8x2 + 4((-4 * 12 + x * 12) + x(-4 + x)) = 32(x + -6)(x + 30)
-800 + 120x + 8x2 + 4((-48 + 12x) + x(-4 + x)) = 32(x + -6)(x + 30)
-800 + 120x + 8x2 + 4(-48 + 12x + (-4 * x + x * x)) = 32(x + -6)(x + 30)
-800 + 120x + 8x2 + 4(-48 + 12x + (-4x + x2)) = 32(x + -6)(x + 30)

Combine like terms: 12x + -4x = 8x
-800 + 120x + 8x2 + 4(-48 + 8x + x2) = 32(x + -6)(x + 30)
-800 + 120x + 8x2 + (-48 * 4 + 8x * 4 + x2 * 4) = 32(x + -6)(x + 30)
-800 + 120x + 8x2 + (-192 + 32x + 4x2) = 32(x + -6)(x + 30)

Reorder the terms:
-800 + -192 + 120x + 32x + 8x2 + 4x2 = 32(x + -6)(x + 30)

Combine like terms: -800 + -192 = -992
-992 + 120x + 32x + 8x2 + 4x2 = 32(x + -6)(x + 30)

Combine like terms: 120x + 32x = 152x
-992 + 152x + 8x2 + 4x2 = 32(x + -6)(x + 30)

Combine like terms: 8x2 + 4x2 = 12x2
-992 + 152x + 12x2 = 32(x + -6)(x + 30)

Reorder the terms:
-992 + 152x + 12x2 = 32(-6 + x)(x + 30)

Reorder the terms:
-992 + 152x + 12x2 = 32(-6 + x)(30 + x)

Multiply (-6 + x) * (30 + x)
-992 + 152x + 12x2 = 32(-6(30 + x) + x(30 + x))
-992 + 152x + 12x2 = 32((30 * -6 + x * -6) + x(30 + x))
-992 + 152x + 12x2 = 32((-180 + -6x) + x(30 + x))
-992 + 152x + 12x2 = 32(-180 + -6x + (30 * x + x * x))
-992 + 152x + 12x2 = 32(-180 + -6x + (30x + x2))

Combine like terms: -6x + 30x = 24x
-992 + 152x + 12x2 = 32(-180 + 24x + x2)
-992 + 152x + 12x2 = (-180 * 32 + 24x * 32 + x2 * 32)
-992 + 152x + 12x2 = (-5760 + 768x + 32x2)

Solving
-992 + 152x + 12x2 = -5760 + 768x + 32x2

Solving for variable 'x'.

Reorder the terms:
-992 + 5760 + 152x + -768x + 12x2 + -32x2 = -5760 + 768x + 32x2 + 5760 + -768x + -32x2

Combine like terms: -992 + 5760 = 4768
4768 + 152x + -768x + 12x2 + -32x2 = -5760 + 768x + 32x2 + 5760 + -768x + -32x2

Combine like terms: 152x + -768x = -616x
4768 + -616x + 12x2 + -32x2 = -5760 + 768x + 32x2 + 5760 + -768x + -32x2

Combine like terms: 12x2 + -32x2 = -20x2
4768 + -616x + -20x2 = -5760 + 768x + 32x2 + 5760 + -768x + -32x2

Reorder the terms:
4768 + -616x + -20x2 = -5760 + 5760 + 768x + -768x + 32x2 + -32x2

Combine like terms: -5760 + 5760 = 0
4768 + -616x + -20x2 = 0 + 768x + -768x + 32x2 + -32x2
4768 + -616x + -20x2 = 768x + -768x + 32x2 + -32x2

Combine like terms: 768x + -768x = 0
4768 + -616x + -20x2 = 0 + 32x2 + -32x2
4768 + -616x + -20x2 = 32x2 + -32x2

Combine like terms: 32x2 + -32x2 = 0
4768 + -616x + -20x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(1192 + -154x + -5x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(1192 + -154x + -5x2)' equal to zero and attempt to solve:

Simplifying
1192 + -154x + -5x2 = 0

Solving
1192 + -154x + -5x2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-238.4 + 30.8x + x2 = 0

Move the constant term to the right:

Add '238.4' to each side of the equation.
-238.4 + 30.8x + 238.4 + x2 = 0 + 238.4

Reorder the terms:
-238.4 + 238.4 + 30.8x + x2 = 0 + 238.4

Combine like terms: -238.4 + 238.4 = 0.0
0.0 + 30.8x + x2 = 0 + 238.4
30.8x + x2 = 0 + 238.4

Combine like terms: 0 + 238.4 = 238.4
30.8x + x2 = 238.4

The x term is 30.8x.  Take half its coefficient (15.4).
Square it (237.16) and add it to both sides.

Add '237.16' to each side of the equation.
30.8x + 237.16 + x2 = 238.4 + 237.16

Reorder the terms:
237.16 + 30.8x + x2 = 238.4 + 237.16

Combine like terms: 238.4 + 237.16 = 475.56
237.16 + 30.8x + x2 = 475.56

Factor a perfect square on the left side:
(x + 15.4)(x + 15.4) = 475.56

Calculate the square root of the right side: 21.807338214

Break this problem into two subproblems by setting 
(x + 15.4) equal to 21.807338214 and -21.807338214.

Subproblem 1
x + 15.4 = 21.807338214

Simplifying
x + 15.4 = 21.807338214

Reorder the terms:
15.4 + x = 21.807338214

Solving
15.4 + x = 21.807338214

Solving for variable 'x'.

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

Add '-15.4' to each side of the equation.
15.4 + -15.4 + x = 21.807338214 + -15.4

Combine like terms: 15.4 + -15.4 = 0.0
0.0 + x = 21.807338214 + -15.4
x = 21.807338214 + -15.4

Combine like terms: 21.807338214 + -15.4 = 6.407338214
x = 6.407338214

Simplifying
x = 6.407338214

Subproblem 2
x + 15.4 = -21.807338214

Simplifying
x + 15.4 = -21.807338214

Reorder the terms:
15.4 + x = -21.807338214

Solving
15.4 + x = -21.807338214

Solving for variable 'x'.

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

Add '-15.4' to each side of the equation.
15.4 + -15.4 + x = -21.807338214 + -15.4

Combine like terms: 15.4 + -15.4 = 0.0
0.0 + x = -21.807338214 + -15.4
x = -21.807338214 + -15.4

Combine like terms: -21.807338214 + -15.4 = -37.207338214
x = -37.207338214

Simplifying
x = -37.207338214

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.407338214, -37.207338214}
Solution
x = {6.407338214, -37.207338214}