2(y-2)=4(y+6)(-2y)

Solution for 2(y-2)=4(y+6)(-2y) equation:

Simplifying
2(y + -2) = 4(y + 6)(-2y)

Reorder the terms:
2(-2 + y) = 4(y + 6)(-2y)
(-2 * 2 + y * 2) = 4(y + 6)(-2y)
(-4 + 2y) = 4(y + 6)(-2y)

Reorder the terms:
-4 + 2y = 4(6 + y)(-2y)

Remove parenthesis around (-2y)
-4 + 2y = 4(6 + y) * -2y

Reorder the terms for easier multiplication:
-4 + 2y = 4 * -2y(6 + y)

Multiply 4 * -2
-4 + 2y = -8y(6 + y)
-4 + 2y = (6 * -8y + y * -8y)
-4 + 2y = (-48y + -8y2)

Solving
-4 + 2y = -48y + -8y2

Solving for variable 'y'.

Combine like terms: 2y + 48y = 50y
-4 + 50y + 8y2 = -48y + -8y2 + 48y + 8y2

Reorder the terms:
-4 + 50y + 8y2 = -48y + 48y + -8y2 + 8y2

Combine like terms: -48y + 48y = 0
-4 + 50y + 8y2 = 0 + -8y2 + 8y2
-4 + 50y + 8y2 = -8y2 + 8y2

Combine like terms: -8y2 + 8y2 = 0
-4 + 50y + 8y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + 25y + 4y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-2 + 25y + 4y2)' equal to zero and attempt to solve:

Simplifying
-2 + 25y + 4y2 = 0

Solving
-2 + 25y + 4y2 = 0

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

Divide each side by '4'.
-0.5 + 6.25y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 6.25y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 6.25y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 6.25y + y2 = 0 + 0.5
6.25y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
6.25y + y2 = 0.5

The y term is 6.25y.  Take half its coefficient (3.125).
Square it (9.765625) and add it to both sides.

Add '9.765625' to each side of the equation.
6.25y + 9.765625 + y2 = 0.5 + 9.765625

Reorder the terms:
9.765625 + 6.25y + y2 = 0.5 + 9.765625

Combine like terms: 0.5 + 9.765625 = 10.265625
9.765625 + 6.25y + y2 = 10.265625

Factor a perfect square on the left side:
(y + 3.125)(y + 3.125) = 10.265625

Calculate the square root of the right side: 3.204001404

Break this problem into two subproblems by setting 
(y + 3.125) equal to 3.204001404 and -3.204001404.

Subproblem 1
y + 3.125 = 3.204001404

Simplifying
y + 3.125 = 3.204001404

Reorder the terms:
3.125 + y = 3.204001404

Solving
3.125 + y = 3.204001404

Solving for variable 'y'.

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

Add '-3.125' to each side of the equation.
3.125 + -3.125 + y = 3.204001404 + -3.125

Combine like terms: 3.125 + -3.125 = 0.000
0.000 + y = 3.204001404 + -3.125
y = 3.204001404 + -3.125

Combine like terms: 3.204001404 + -3.125 = 0.079001404
y = 0.079001404

Simplifying
y = 0.079001404

Subproblem 2
y + 3.125 = -3.204001404

Simplifying
y + 3.125 = -3.204001404

Reorder the terms:
3.125 + y = -3.204001404

Solving
3.125 + y = -3.204001404

Solving for variable 'y'.

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

Add '-3.125' to each side of the equation.
3.125 + -3.125 + y = -3.204001404 + -3.125

Combine like terms: 3.125 + -3.125 = 0.000
0.000 + y = -3.204001404 + -3.125
y = -3.204001404 + -3.125

Combine like terms: -3.204001404 + -3.125 = -6.329001404
y = -6.329001404

Simplifying
y = -6.329001404

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.079001404, -6.329001404}
Solution
y = {0.079001404, -6.329001404}