4y(2y+6)+4y+6(3y+9)=9y+19(y+3)

Solution for 4y(2y+6)+4y+6(3y+9)=9y+19(y+3) equation:

Simplifying
4y(2y + 6) + 4y + 6(3y + 9) = 9y + 19(y + 3)

Reorder the terms:
4y(6 + 2y) + 4y + 6(3y + 9) = 9y + 19(y + 3)
(6 * 4y + 2y * 4y) + 4y + 6(3y + 9) = 9y + 19(y + 3)
(24y + 8y2) + 4y + 6(3y + 9) = 9y + 19(y + 3)

Reorder the terms:
24y + 8y2 + 4y + 6(9 + 3y) = 9y + 19(y + 3)
24y + 8y2 + 4y + (9 * 6 + 3y * 6) = 9y + 19(y + 3)
24y + 8y2 + 4y + (54 + 18y) = 9y + 19(y + 3)

Reorder the terms:
54 + 24y + 4y + 18y + 8y2 = 9y + 19(y + 3)

Combine like terms: 24y + 4y = 28y
54 + 28y + 18y + 8y2 = 9y + 19(y + 3)

Combine like terms: 28y + 18y = 46y
54 + 46y + 8y2 = 9y + 19(y + 3)

Reorder the terms:
54 + 46y + 8y2 = 9y + 19(3 + y)
54 + 46y + 8y2 = 9y + (3 * 19 + y * 19)
54 + 46y + 8y2 = 9y + (57 + 19y)

Reorder the terms:
54 + 46y + 8y2 = 57 + 9y + 19y

Combine like terms: 9y + 19y = 28y
54 + 46y + 8y2 = 57 + 28y

Solving
54 + 46y + 8y2 = 57 + 28y

Solving for variable 'y'.

Reorder the terms:
54 + -57 + 46y + -28y + 8y2 = 57 + 28y + -57 + -28y

Combine like terms: 54 + -57 = -3
-3 + 46y + -28y + 8y2 = 57 + 28y + -57 + -28y

Combine like terms: 46y + -28y = 18y
-3 + 18y + 8y2 = 57 + 28y + -57 + -28y

Reorder the terms:
-3 + 18y + 8y2 = 57 + -57 + 28y + -28y

Combine like terms: 57 + -57 = 0
-3 + 18y + 8y2 = 0 + 28y + -28y
-3 + 18y + 8y2 = 28y + -28y

Combine like terms: 28y + -28y = 0
-3 + 18y + 8y2 = 0

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

Divide each side by '8'.
-0.375 + 2.25y + y2 = 0

Move the constant term to the right:

Add '0.375' to each side of the equation.
-0.375 + 2.25y + 0.375 + y2 = 0 + 0.375

Reorder the terms:
-0.375 + 0.375 + 2.25y + y2 = 0 + 0.375

Combine like terms: -0.375 + 0.375 = 0.000
0.000 + 2.25y + y2 = 0 + 0.375
2.25y + y2 = 0 + 0.375

Combine like terms: 0 + 0.375 = 0.375
2.25y + y2 = 0.375

The y term is 2.25y.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25y + 1.265625 + y2 = 0.375 + 1.265625

Reorder the terms:
1.265625 + 2.25y + y2 = 0.375 + 1.265625

Combine like terms: 0.375 + 1.265625 = 1.640625
1.265625 + 2.25y + y2 = 1.640625

Factor a perfect square on the left side:
(y + 1.125)(y + 1.125) = 1.640625

Calculate the square root of the right side: 1.280868846

Break this problem into two subproblems by setting 
(y + 1.125) equal to 1.280868846 and -1.280868846.

Subproblem 1
y + 1.125 = 1.280868846

Simplifying
y + 1.125 = 1.280868846

Reorder the terms:
1.125 + y = 1.280868846

Solving
1.125 + y = 1.280868846

Solving for variable 'y'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + y = 1.280868846 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + y = 1.280868846 + -1.125
y = 1.280868846 + -1.125

Combine like terms: 1.280868846 + -1.125 = 0.155868846
y = 0.155868846

Simplifying
y = 0.155868846

Subproblem 2
y + 1.125 = -1.280868846

Simplifying
y + 1.125 = -1.280868846

Reorder the terms:
1.125 + y = -1.280868846

Solving
1.125 + y = -1.280868846

Solving for variable 'y'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + y = -1.280868846 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + y = -1.280868846 + -1.125
y = -1.280868846 + -1.125

Combine like terms: -1.280868846 + -1.125 = -2.405868846
y = -2.405868846

Simplifying
y = -2.405868846

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.155868846, -2.405868846}