2(y+2)y=19-(2y+3)

Solution for 2(y+2)y=19-(2y+3) equation:

Simplifying
2(y + 2) * y = 19 + -1(2y + 3)

Reorder the terms:
2(2 + y) * y = 19 + -1(2y + 3)

Reorder the terms for easier multiplication:
2y(2 + y) = 19 + -1(2y + 3)
(2 * 2y + y * 2y) = 19 + -1(2y + 3)
(4y + 2y2) = 19 + -1(2y + 3)

Reorder the terms:
4y + 2y2 = 19 + -1(3 + 2y)
4y + 2y2 = 19 + (3 * -1 + 2y * -1)
4y + 2y2 = 19 + (-3 + -2y)

Combine like terms: 19 + -3 = 16
4y + 2y2 = 16 + -2y

Solving
4y + 2y2 = 16 + -2y

Solving for variable 'y'.

Reorder the terms:
-16 + 4y + 2y + 2y2 = 16 + -2y + -16 + 2y

Combine like terms: 4y + 2y = 6y
-16 + 6y + 2y2 = 16 + -2y + -16 + 2y

Reorder the terms:
-16 + 6y + 2y2 = 16 + -16 + -2y + 2y

Combine like terms: 16 + -16 = 0
-16 + 6y + 2y2 = 0 + -2y + 2y
-16 + 6y + 2y2 = -2y + 2y

Combine like terms: -2y + 2y = 0
-16 + 6y + 2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-8 + 3y + y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-8 + 3y + y2)' equal to zero and attempt to solve:

Simplifying
-8 + 3y + y2 = 0

Solving
-8 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 3y + 8 + y2 = 0 + 8

Reorder the terms:
-8 + 8 + 3y + y2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 3y + y2 = 0 + 8
3y + y2 = 0 + 8

Combine like terms: 0 + 8 = 8
3y + y2 = 8

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 8 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 8 + 2.25

Combine like terms: 8 + 2.25 = 10.25
2.25 + 3y + y2 = 10.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(y + 1.5) equal to 3.201562119 and -3.201562119.

Subproblem 1
y + 1.5 = 3.201562119

Simplifying
y + 1.5 = 3.201562119

Reorder the terms:
1.5 + y = 3.201562119

Solving
1.5 + y = 3.201562119

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 3.201562119 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 3.201562119 + -1.5
y = 3.201562119 + -1.5

Combine like terms: 3.201562119 + -1.5 = 1.701562119
y = 1.701562119

Simplifying
y = 1.701562119

Subproblem 2
y + 1.5 = -3.201562119

Simplifying
y + 1.5 = -3.201562119

Reorder the terms:
1.5 + y = -3.201562119

Solving
1.5 + y = -3.201562119

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -3.201562119 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -3.201562119 + -1.5
y = -3.201562119 + -1.5

Combine like terms: -3.201562119 + -1.5 = -4.701562119
y = -4.701562119

Simplifying
y = -4.701562119

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.701562119, -4.701562119}
Solution
y = {1.701562119, -4.701562119}