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

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

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

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

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

Multiply (3 + y) * (2 + y)
(3(2 + y) + y(2 + y)) + (y + 3)(y + -6) = (y + -6)(y + 2)
((2 * 3 + y * 3) + y(2 + y)) + (y + 3)(y + -6) = (y + -6)(y + 2)
((6 + 3y) + y(2 + y)) + (y + 3)(y + -6) = (y + -6)(y + 2)
(6 + 3y + (2 * y + y * y)) + (y + 3)(y + -6) = (y + -6)(y + 2)
(6 + 3y + (2y + y2)) + (y + 3)(y + -6) = (y + -6)(y + 2)

Combine like terms: 3y + 2y = 5y
(6 + 5y + y2) + (y + 3)(y + -6) = (y + -6)(y + 2)

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

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

Multiply (3 + y) * (-6 + y)
6 + 5y + y2 + (3(-6 + y) + y(-6 + y)) = (y + -6)(y + 2)
6 + 5y + y2 + ((-6 * 3 + y * 3) + y(-6 + y)) = (y + -6)(y + 2)
6 + 5y + y2 + ((-18 + 3y) + y(-6 + y)) = (y + -6)(y + 2)
6 + 5y + y2 + (-18 + 3y + (-6 * y + y * y)) = (y + -6)(y + 2)
6 + 5y + y2 + (-18 + 3y + (-6y + y2)) = (y + -6)(y + 2)

Combine like terms: 3y + -6y = -3y
6 + 5y + y2 + (-18 + -3y + y2) = (y + -6)(y + 2)

Reorder the terms:
6 + -18 + 5y + -3y + y2 + y2 = (y + -6)(y + 2)

Combine like terms: 6 + -18 = -12
-12 + 5y + -3y + y2 + y2 = (y + -6)(y + 2)

Combine like terms: 5y + -3y = 2y
-12 + 2y + y2 + y2 = (y + -6)(y + 2)

Combine like terms: y2 + y2 = 2y2
-12 + 2y + 2y2 = (y + -6)(y + 2)

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

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

Multiply (-6 + y) * (2 + y)
-12 + 2y + 2y2 = (-6(2 + y) + y(2 + y))
-12 + 2y + 2y2 = ((2 * -6 + y * -6) + y(2 + y))
-12 + 2y + 2y2 = ((-12 + -6y) + y(2 + y))
-12 + 2y + 2y2 = (-12 + -6y + (2 * y + y * y))
-12 + 2y + 2y2 = (-12 + -6y + (2y + y2))

Combine like terms: -6y + 2y = -4y
-12 + 2y + 2y2 = (-12 + -4y + y2)

Add '12' to each side of the equation.
-12 + 2y + 12 + 2y2 = -12 + -4y + 12 + y2

Reorder the terms:
-12 + 12 + 2y + 2y2 = -12 + -4y + 12 + y2

Combine like terms: -12 + 12 = 0
0 + 2y + 2y2 = -12 + -4y + 12 + y2
2y + 2y2 = -12 + -4y + 12 + y2

Reorder the terms:
2y + 2y2 = -12 + 12 + -4y + y2

Combine like terms: -12 + 12 = 0
2y + 2y2 = 0 + -4y + y2
2y + 2y2 = -4y + y2

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

Solving for variable 'y'.

Reorder the terms:
2y + 4y + 2y2 + -1y2 = -4y + y2 + 4y + -1y2

Combine like terms: 2y + 4y = 6y
6y + 2y2 + -1y2 = -4y + y2 + 4y + -1y2

Combine like terms: 2y2 + -1y2 = 1y2
6y + 1y2 = -4y + y2 + 4y + -1y2

Reorder the terms:
6y + 1y2 = -4y + 4y + y2 + -1y2

Combine like terms: -4y + 4y = 0
6y + 1y2 = 0 + y2 + -1y2
6y + 1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
6y + 1y2 = 0

Factor out the Greatest Common Factor (GCF), 'y'.
y(6 + y) = 0

Subproblem 1
Set the factor 'y' equal to zero and attempt to solve:

Simplifying
y = 0

Solving
y = 0

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

Simplifying
y = 0
Subproblem 2
Set the factor '(6 + y)' equal to zero and attempt to solve:

Simplifying
6 + y = 0

Solving
6 + y = 0

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

Add '-6' to each side of the equation.
6 + -6 + y = 0 + -6

Combine like terms: 6 + -6 = 0
0 + y = 0 + -6
y = 0 + -6

Combine like terms: 0 + -6 = -6
y = -6

Simplifying
y = -6
Solution
y = {0, -6}