y(y+1)-4=2(1+y)-(y-3)(y+2)

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Solution for y(y+1)-4=2(1+y)-(y-3)(y+2) equation: 


Simplifying
y(y + 1) + -4 = 2(1 + y) + -1(y + -3)(y + 2)

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

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

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

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

Multiply (-3 + y) * (2 + y)
-4 + 1y + y2 = 2 + 2y + -1(-3(2 + y) + y(2 + y))
-4 + 1y + y2 = 2 + 2y + -1((2 * -3 + y * -3) + y(2 + y))
-4 + 1y + y2 = 2 + 2y + -1((-6 + -3y) + y(2 + y))
-4 + 1y + y2 = 2 + 2y + -1(-6 + -3y + (2 * y + y * y))
-4 + 1y + y2 = 2 + 2y + -1(-6 + -3y + (2y + y2))

Combine like terms: -3y + 2y = -1y
-4 + 1y + y2 = 2 + 2y + -1(-6 + -1y + y2)
-4 + 1y + y2 = 2 + 2y + (-6 * -1 + -1y * -1 + y2 * -1)
-4 + 1y + y2 = 2 + 2y + (6 + 1y + -1y2)

Reorder the terms:
-4 + 1y + y2 = 2 + 6 + 2y + 1y + -1y2

Combine like terms: 2 + 6 = 8
-4 + 1y + y2 = 8 + 2y + 1y + -1y2

Combine like terms: 2y + 1y = 3y
-4 + 1y + y2 = 8 + 3y + -1y2

Solving
-4 + 1y + y2 = 8 + 3y + -1y2

Solving for variable 'y'.

Reorder the terms:
-4 + -8 + 1y + -3y + y2 + y2 = 8 + 3y + -1y2 + -8 + -3y + y2

Combine like terms: -4 + -8 = -12
-12 + 1y + -3y + y2 + y2 = 8 + 3y + -1y2 + -8 + -3y + y2

Combine like terms: 1y + -3y = -2y
-12 + -2y + y2 + y2 = 8 + 3y + -1y2 + -8 + -3y + y2

Combine like terms: y2 + y2 = 2y2
-12 + -2y + 2y2 = 8 + 3y + -1y2 + -8 + -3y + y2

Reorder the terms:
-12 + -2y + 2y2 = 8 + -8 + 3y + -3y + -1y2 + y2

Combine like terms: 8 + -8 = 0
-12 + -2y + 2y2 = 0 + 3y + -3y + -1y2 + y2
-12 + -2y + 2y2 = 3y + -3y + -1y2 + y2

Combine like terms: 3y + -3y = 0
-12 + -2y + 2y2 = 0 + -1y2 + y2
-12 + -2y + 2y2 = -1y2 + y2

Combine like terms: -1y2 + y2 = 0
-12 + -2y + 2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + -1y + y2) = 0

Factor a trinomial.
2((-2 + -1y)(3 + -1y)) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-2 + -1y)' equal to zero and attempt to solve:

Simplifying
-2 + -1y = 0

Solving
-2 + -1y = 0

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

Add '2' to each side of the equation.
-2 + 2 + -1y = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1y = 0 + 2
-1y = 0 + 2

Combine like terms: 0 + 2 = 2
-1y = 2

Divide each side by '-1'.
y = -2

Simplifying
y = -2

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

Simplifying
3 + -1y = 0

Solving
3 + -1y = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1y = 0 + -3

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

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

Divide each side by '-1'.
y = 3

Simplifying
y = 3
Solution
y = {-2, 3}

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