8(x-6)(x+4)+4(x+2)(x-8)=32(x-16)(x-8)

Solution for 8(x-6)(x+4)+4(x+2)(x-8)=32(x-16)(x-8) equation:

Simplifying
8(x + -6)(x + 4) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)

Reorder the terms:
8(-6 + x)(x + 4) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)

Reorder the terms:
8(-6 + x)(4 + x) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)

Multiply (-6 + x) * (4 + x)
8(-6(4 + x) + x(4 + x)) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)
8((4 * -6 + x * -6) + x(4 + x)) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)
8((-24 + -6x) + x(4 + x)) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)
8(-24 + -6x + (4 * x + x * x)) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)
8(-24 + -6x + (4x + x2)) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)

Combine like terms: -6x + 4x = -2x
8(-24 + -2x + x2) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)
(-24 * 8 + -2x * 8 + x2 * 8) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)
(-192 + -16x + 8x2) + 4(x + 2)(x + -8) = 32(x + -16)(x + -8)

Reorder the terms:
-192 + -16x + 8x2 + 4(2 + x)(x + -8) = 32(x + -16)(x + -8)

Reorder the terms:
-192 + -16x + 8x2 + 4(2 + x)(-8 + x) = 32(x + -16)(x + -8)

Multiply (2 + x) * (-8 + x)
-192 + -16x + 8x2 + 4(2(-8 + x) + x(-8 + x)) = 32(x + -16)(x + -8)
-192 + -16x + 8x2 + 4((-8 * 2 + x * 2) + x(-8 + x)) = 32(x + -16)(x + -8)
-192 + -16x + 8x2 + 4((-16 + 2x) + x(-8 + x)) = 32(x + -16)(x + -8)
-192 + -16x + 8x2 + 4(-16 + 2x + (-8 * x + x * x)) = 32(x + -16)(x + -8)
-192 + -16x + 8x2 + 4(-16 + 2x + (-8x + x2)) = 32(x + -16)(x + -8)

Combine like terms: 2x + -8x = -6x
-192 + -16x + 8x2 + 4(-16 + -6x + x2) = 32(x + -16)(x + -8)
-192 + -16x + 8x2 + (-16 * 4 + -6x * 4 + x2 * 4) = 32(x + -16)(x + -8)
-192 + -16x + 8x2 + (-64 + -24x + 4x2) = 32(x + -16)(x + -8)

Reorder the terms:
-192 + -64 + -16x + -24x + 8x2 + 4x2 = 32(x + -16)(x + -8)

Combine like terms: -192 + -64 = -256
-256 + -16x + -24x + 8x2 + 4x2 = 32(x + -16)(x + -8)

Combine like terms: -16x + -24x = -40x
-256 + -40x + 8x2 + 4x2 = 32(x + -16)(x + -8)

Combine like terms: 8x2 + 4x2 = 12x2
-256 + -40x + 12x2 = 32(x + -16)(x + -8)

Reorder the terms:
-256 + -40x + 12x2 = 32(-16 + x)(x + -8)

Reorder the terms:
-256 + -40x + 12x2 = 32(-16 + x)(-8 + x)

Multiply (-16 + x) * (-8 + x)
-256 + -40x + 12x2 = 32(-16(-8 + x) + x(-8 + x))
-256 + -40x + 12x2 = 32((-8 * -16 + x * -16) + x(-8 + x))
-256 + -40x + 12x2 = 32((128 + -16x) + x(-8 + x))
-256 + -40x + 12x2 = 32(128 + -16x + (-8 * x + x * x))
-256 + -40x + 12x2 = 32(128 + -16x + (-8x + x2))

Combine like terms: -16x + -8x = -24x
-256 + -40x + 12x2 = 32(128 + -24x + x2)
-256 + -40x + 12x2 = (128 * 32 + -24x * 32 + x2 * 32)
-256 + -40x + 12x2 = (4096 + -768x + 32x2)

Solving
-256 + -40x + 12x2 = 4096 + -768x + 32x2

Solving for variable 'x'.

Reorder the terms:
-256 + -4096 + -40x + 768x + 12x2 + -32x2 = 4096 + -768x + 32x2 + -4096 + 768x + -32x2

Combine like terms: -256 + -4096 = -4352
-4352 + -40x + 768x + 12x2 + -32x2 = 4096 + -768x + 32x2 + -4096 + 768x + -32x2

Combine like terms: -40x + 768x = 728x
-4352 + 728x + 12x2 + -32x2 = 4096 + -768x + 32x2 + -4096 + 768x + -32x2

Combine like terms: 12x2 + -32x2 = -20x2
-4352 + 728x + -20x2 = 4096 + -768x + 32x2 + -4096 + 768x + -32x2

Reorder the terms:
-4352 + 728x + -20x2 = 4096 + -4096 + -768x + 768x + 32x2 + -32x2

Combine like terms: 4096 + -4096 = 0
-4352 + 728x + -20x2 = 0 + -768x + 768x + 32x2 + -32x2
-4352 + 728x + -20x2 = -768x + 768x + 32x2 + -32x2

Combine like terms: -768x + 768x = 0
-4352 + 728x + -20x2 = 0 + 32x2 + -32x2
-4352 + 728x + -20x2 = 32x2 + -32x2

Combine like terms: 32x2 + -32x2 = 0
-4352 + 728x + -20x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-1088 + 182x + -5x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-1088 + 182x + -5x2)' equal to zero and attempt to solve:

Simplifying
-1088 + 182x + -5x2 = 0

Solving
-1088 + 182x + -5x2 = 0

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

Divide each side by '-5'.
217.6 + -36.4x + x2 = 0

Move the constant term to the right:

Add '-217.6' to each side of the equation.
217.6 + -36.4x + -217.6 + x2 = 0 + -217.6

Reorder the terms:
217.6 + -217.6 + -36.4x + x2 = 0 + -217.6

Combine like terms: 217.6 + -217.6 = 0.0
0.0 + -36.4x + x2 = 0 + -217.6
-36.4x + x2 = 0 + -217.6

Combine like terms: 0 + -217.6 = -217.6
-36.4x + x2 = -217.6

The x term is -36.4x.  Take half its coefficient (-18.2).
Square it (331.24) and add it to both sides.

Add '331.24' to each side of the equation.
-36.4x + 331.24 + x2 = -217.6 + 331.24

Reorder the terms:
331.24 + -36.4x + x2 = -217.6 + 331.24

Combine like terms: -217.6 + 331.24 = 113.64
331.24 + -36.4x + x2 = 113.64

Factor a perfect square on the left side:
(x + -18.2)(x + -18.2) = 113.64

Calculate the square root of the right side: 10.660206377

Break this problem into two subproblems by setting 
(x + -18.2) equal to 10.660206377 and -10.660206377.

Subproblem 1
x + -18.2 = 10.660206377

Simplifying
x + -18.2 = 10.660206377

Reorder the terms:
-18.2 + x = 10.660206377

Solving
-18.2 + x = 10.660206377

Solving for variable 'x'.

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

Add '18.2' to each side of the equation.
-18.2 + 18.2 + x = 10.660206377 + 18.2

Combine like terms: -18.2 + 18.2 = 0.0
0.0 + x = 10.660206377 + 18.2
x = 10.660206377 + 18.2

Combine like terms: 10.660206377 + 18.2 = 28.860206377
x = 28.860206377

Simplifying
x = 28.860206377

Subproblem 2
x + -18.2 = -10.660206377

Simplifying
x + -18.2 = -10.660206377

Reorder the terms:
-18.2 + x = -10.660206377

Solving
-18.2 + x = -10.660206377

Solving for variable 'x'.

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

Add '18.2' to each side of the equation.
-18.2 + 18.2 + x = -10.660206377 + 18.2

Combine like terms: -18.2 + 18.2 = 0.0
0.0 + x = -10.660206377 + 18.2
x = -10.660206377 + 18.2

Combine like terms: -10.660206377 + 18.2 = 7.539793623
x = 7.539793623

Simplifying
x = 7.539793623

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {28.860206377, 7.539793623}
Solution
x = {28.860206377, 7.539793623}