2y(y+3)=(4y-2)(y+5)+y(y+5)

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Solution for 2y(y+3)=(4y-2)(y+5)+y(y+5) equation: 


Simplifying
2y(y + 3) = (4y + -2)(y + 5) + y(y + 5)

Reorder the terms:
2y(3 + y) = (4y + -2)(y + 5) + y(y + 5)
(3 * 2y + y * 2y) = (4y + -2)(y + 5) + y(y + 5)
(6y + 2y2) = (4y + -2)(y + 5) + y(y + 5)

Reorder the terms:
6y + 2y2 = (-2 + 4y)(y + 5) + y(y + 5)

Reorder the terms:
6y + 2y2 = (-2 + 4y)(5 + y) + y(y + 5)

Multiply (-2 + 4y) * (5 + y)
6y + 2y2 = (-2(5 + y) + 4y * (5 + y)) + y(y + 5)
6y + 2y2 = ((5 * -2 + y * -2) + 4y * (5 + y)) + y(y + 5)
6y + 2y2 = ((-10 + -2y) + 4y * (5 + y)) + y(y + 5)
6y + 2y2 = (-10 + -2y + (5 * 4y + y * 4y)) + y(y + 5)
6y + 2y2 = (-10 + -2y + (20y + 4y2)) + y(y + 5)

Combine like terms: -2y + 20y = 18y
6y + 2y2 = (-10 + 18y + 4y2) + y(y + 5)

Reorder the terms:
6y + 2y2 = -10 + 18y + 4y2 + y(5 + y)
6y + 2y2 = -10 + 18y + 4y2 + (5 * y + y * y)
6y + 2y2 = -10 + 18y + 4y2 + (5y + y2)

Reorder the terms:
6y + 2y2 = -10 + 18y + 5y + 4y2 + y2

Combine like terms: 18y + 5y = 23y
6y + 2y2 = -10 + 23y + 4y2 + y2

Combine like terms: 4y2 + y2 = 5y2
6y + 2y2 = -10 + 23y + 5y2

Solving
6y + 2y2 = -10 + 23y + 5y2

Solving for variable 'y'.

Reorder the terms:
10 + 6y + -23y + 2y2 + -5y2 = -10 + 23y + 5y2 + 10 + -23y + -5y2

Combine like terms: 6y + -23y = -17y
10 + -17y + 2y2 + -5y2 = -10 + 23y + 5y2 + 10 + -23y + -5y2

Combine like terms: 2y2 + -5y2 = -3y2
10 + -17y + -3y2 = -10 + 23y + 5y2 + 10 + -23y + -5y2

Reorder the terms:
10 + -17y + -3y2 = -10 + 10 + 23y + -23y + 5y2 + -5y2

Combine like terms: -10 + 10 = 0
10 + -17y + -3y2 = 0 + 23y + -23y + 5y2 + -5y2
10 + -17y + -3y2 = 23y + -23y + 5y2 + -5y2

Combine like terms: 23y + -23y = 0
10 + -17y + -3y2 = 0 + 5y2 + -5y2
10 + -17y + -3y2 = 5y2 + -5y2

Combine like terms: 5y2 + -5y2 = 0
10 + -17y + -3y2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-3.333333333 + 5.666666667y + y2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + 5.666666667y + 3.333333333 + y2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + 5.666666667y + y2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + 5.666666667y + y2 = 0 + 3.333333333
5.666666667y + y2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
5.666666667y + y2 = 3.333333333

The y term is 5.666666667y.  Take half its coefficient (2.833333334).
Square it (8.027777782) and add it to both sides.

Add '8.027777782' to each side of the equation.
5.666666667y + 8.027777782 + y2 = 3.333333333 + 8.027777782

Reorder the terms:
8.027777782 + 5.666666667y + y2 = 3.333333333 + 8.027777782

Combine like terms: 3.333333333 + 8.027777782 = 11.361111115
8.027777782 + 5.666666667y + y2 = 11.361111115

Factor a perfect square on the left side:
(y + 2.833333334)(y + 2.833333334) = 11.361111115

Calculate the square root of the right side: 3.370624737

Break this problem into two subproblems by setting 
(y + 2.833333334) equal to 3.370624737 and -3.370624737.


Subproblem 1
y + 2.833333334 = 3.370624737

Simplifying
y + 2.833333334 = 3.370624737

Reorder the terms:
2.833333334 + y = 3.370624737

Solving
2.833333334 + y = 3.370624737

Solving for variable 'y'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + y = 3.370624737 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + y = 3.370624737 + -2.833333334
y = 3.370624737 + -2.833333334

Combine like terms: 3.370624737 + -2.833333334 = 0.537291403
y = 0.537291403

Simplifying
y = 0.537291403


Subproblem 2
y + 2.833333334 = -3.370624737

Simplifying
y + 2.833333334 = -3.370624737

Reorder the terms:
2.833333334 + y = -3.370624737

Solving
2.833333334 + y = -3.370624737

Solving for variable 'y'.

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

Add '-2.833333334' to each side of the equation.
2.833333334 + -2.833333334 + y = -3.370624737 + -2.833333334

Combine like terms: 2.833333334 + -2.833333334 = 0.000000000
0.000000000 + y = -3.370624737 + -2.833333334
y = -3.370624737 + -2.833333334

Combine like terms: -3.370624737 + -2.833333334 = -6.203958071
y = -6.203958071

Simplifying
y = -6.203958071


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.537291403, -6.203958071}

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