-2y(y+6)=y+3+2y

Solution for -2y(y+6)=y+3+2y equation:

Simplifying
-2y(y + 6) = y + 3 + 2y

Reorder the terms:
-2y(6 + y) = y + 3 + 2y
(6 * -2y + y * -2y) = y + 3 + 2y
(-12y + -2y2) = y + 3 + 2y

Reorder the terms:
-12y + -2y2 = 3 + y + 2y

Combine like terms: y + 2y = 3y
-12y + -2y2 = 3 + 3y

Solving
-12y + -2y2 = 3 + 3y

Solving for variable 'y'.

Reorder the terms:
-3 + -12y + -3y + -2y2 = 3 + 3y + -3 + -3y

Combine like terms: -12y + -3y = -15y
-3 + -15y + -2y2 = 3 + 3y + -3 + -3y

Reorder the terms:
-3 + -15y + -2y2 = 3 + -3 + 3y + -3y

Combine like terms: 3 + -3 = 0
-3 + -15y + -2y2 = 0 + 3y + -3y
-3 + -15y + -2y2 = 3y + -3y

Combine like terms: 3y + -3y = 0
-3 + -15y + -2y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(3 + 15y + 2y2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(3 + 15y + 2y2)' equal to zero and attempt to solve:

Simplifying
3 + 15y + 2y2 = 0

Solving
3 + 15y + 2y2 = 0

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

Divide each side by '2'.
1.5 + 7.5y + y2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 7.5y + -1.5 + y2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 7.5y + y2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 7.5y + y2 = 0 + -1.5
7.5y + y2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
7.5y + y2 = -1.5

The y term is 7.5y.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5y + 14.0625 + y2 = -1.5 + 14.0625

Reorder the terms:
14.0625 + 7.5y + y2 = -1.5 + 14.0625

Combine like terms: -1.5 + 14.0625 = 12.5625
14.0625 + 7.5y + y2 = 12.5625

Factor a perfect square on the left side:
(y + 3.75)(y + 3.75) = 12.5625

Calculate the square root of the right side: 3.54436172

Break this problem into two subproblems by setting 
(y + 3.75) equal to 3.54436172 and -3.54436172.

Subproblem 1
y + 3.75 = 3.54436172

Simplifying
y + 3.75 = 3.54436172

Reorder the terms:
3.75 + y = 3.54436172

Solving
3.75 + y = 3.54436172

Solving for variable 'y'.

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

Add '-3.75' to each side of the equation.
3.75 + -3.75 + y = 3.54436172 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + y = 3.54436172 + -3.75
y = 3.54436172 + -3.75

Combine like terms: 3.54436172 + -3.75 = -0.20563828
y = -0.20563828

Simplifying
y = -0.20563828

Subproblem 2
y + 3.75 = -3.54436172

Simplifying
y + 3.75 = -3.54436172

Reorder the terms:
3.75 + y = -3.54436172

Solving
3.75 + y = -3.54436172

Solving for variable 'y'.

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

Add '-3.75' to each side of the equation.
3.75 + -3.75 + y = -3.54436172 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + y = -3.54436172 + -3.75
y = -3.54436172 + -3.75

Combine like terms: -3.54436172 + -3.75 = -7.29436172
y = -7.29436172

Simplifying
y = -7.29436172

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.20563828, -7.29436172}
Solution
y = {-0.20563828, -7.29436172}