3y(y+1)+(y-3)=y+9

Solution for 3y(y+1)+(y-3)=y+9 equation:

Simplifying
3y(y + 1) + (y + -3) = y + 9

Reorder the terms:
3y(1 + y) + (y + -3) = y + 9
(1 * 3y + y * 3y) + (y + -3) = y + 9
(3y + 3y2) + (y + -3) = y + 9

Reorder the terms:
3y + 3y2 + (-3 + y) = y + 9

Remove parenthesis around (-3 + y)
3y + 3y2 + -3 + y = y + 9

Reorder the terms:
-3 + 3y + y + 3y2 = y + 9

Combine like terms: 3y + y = 4y
-3 + 4y + 3y2 = y + 9

Reorder the terms:
-3 + 4y + 3y2 = 9 + y

Solving
-3 + 4y + 3y2 = 9 + y

Solving for variable 'y'.

Reorder the terms:
-3 + -9 + 4y + -1y + 3y2 = 9 + y + -9 + -1y

Combine like terms: -3 + -9 = -12
-12 + 4y + -1y + 3y2 = 9 + y + -9 + -1y

Combine like terms: 4y + -1y = 3y
-12 + 3y + 3y2 = 9 + y + -9 + -1y

Reorder the terms:
-12 + 3y + 3y2 = 9 + -9 + y + -1y

Combine like terms: 9 + -9 = 0
-12 + 3y + 3y2 = 0 + y + -1y
-12 + 3y + 3y2 = y + -1y

Combine like terms: y + -1y = 0
-12 + 3y + 3y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-4 + y + y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-4 + y + y2)' equal to zero and attempt to solve:

Simplifying
-4 + y + y2 = 0

Solving
-4 + y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + y + y2 = 0 + 4

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

Combine like terms: 0 + 4 = 4
y + y2 = 4

The y term is y.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
y + 0.25 + y2 = 4 + 0.25

Reorder the terms:
0.25 + y + y2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + y + y2 = 4.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(y + 0.5) equal to 2.061552813 and -2.061552813.

Subproblem 1
y + 0.5 = 2.061552813

Simplifying
y + 0.5 = 2.061552813

Reorder the terms:
0.5 + y = 2.061552813

Solving
0.5 + y = 2.061552813

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = 2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = 2.061552813 + -0.5
y = 2.061552813 + -0.5

Combine like terms: 2.061552813 + -0.5 = 1.561552813
y = 1.561552813

Simplifying
y = 1.561552813

Subproblem 2
y + 0.5 = -2.061552813

Simplifying
y + 0.5 = -2.061552813

Reorder the terms:
0.5 + y = -2.061552813

Solving
0.5 + y = -2.061552813

Solving for variable 'y'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + y = -2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + y = -2.061552813 + -0.5
y = -2.061552813 + -0.5

Combine like terms: -2.061552813 + -0.5 = -2.561552813
y = -2.561552813

Simplifying
y = -2.561552813

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.561552813, -2.561552813}
Solution
y = {1.561552813, -2.561552813}