(y+3)(y+5)=(y+1)(y+6)+(y+6)(y+5)

Solution for (y+3)(y+5)=(y+1)(y+6)+(y+6)(y+5) equation:

Simplifying
(y + 3)(y + 5) = (y + 1)(y + 6) + (y + 6)(y + 5)

Reorder the terms:
(3 + y)(y + 5) = (y + 1)(y + 6) + (y + 6)(y + 5)

Reorder the terms:
(3 + y)(5 + y) = (y + 1)(y + 6) + (y + 6)(y + 5)

Multiply (3 + y) * (5 + y)
(3(5 + y) + y(5 + y)) = (y + 1)(y + 6) + (y + 6)(y + 5)
((5 * 3 + y * 3) + y(5 + y)) = (y + 1)(y + 6) + (y + 6)(y + 5)
((15 + 3y) + y(5 + y)) = (y + 1)(y + 6) + (y + 6)(y + 5)
(15 + 3y + (5 * y + y * y)) = (y + 1)(y + 6) + (y + 6)(y + 5)
(15 + 3y + (5y + y2)) = (y + 1)(y + 6) + (y + 6)(y + 5)

Combine like terms: 3y + 5y = 8y
(15 + 8y + y2) = (y + 1)(y + 6) + (y + 6)(y + 5)

Reorder the terms:
15 + 8y + y2 = (1 + y)(y + 6) + (y + 6)(y + 5)

Reorder the terms:
15 + 8y + y2 = (1 + y)(6 + y) + (y + 6)(y + 5)

Multiply (1 + y) * (6 + y)
15 + 8y + y2 = (1(6 + y) + y(6 + y)) + (y + 6)(y + 5)
15 + 8y + y2 = ((6 * 1 + y * 1) + y(6 + y)) + (y + 6)(y + 5)
15 + 8y + y2 = ((6 + 1y) + y(6 + y)) + (y + 6)(y + 5)
15 + 8y + y2 = (6 + 1y + (6 * y + y * y)) + (y + 6)(y + 5)
15 + 8y + y2 = (6 + 1y + (6y + y2)) + (y + 6)(y + 5)

Combine like terms: 1y + 6y = 7y
15 + 8y + y2 = (6 + 7y + y2) + (y + 6)(y + 5)

Reorder the terms:
15 + 8y + y2 = 6 + 7y + y2 + (6 + y)(y + 5)

Reorder the terms:
15 + 8y + y2 = 6 + 7y + y2 + (6 + y)(5 + y)

Multiply (6 + y) * (5 + y)
15 + 8y + y2 = 6 + 7y + y2 + (6(5 + y) + y(5 + y))
15 + 8y + y2 = 6 + 7y + y2 + ((5 * 6 + y * 6) + y(5 + y))
15 + 8y + y2 = 6 + 7y + y2 + ((30 + 6y) + y(5 + y))
15 + 8y + y2 = 6 + 7y + y2 + (30 + 6y + (5 * y + y * y))
15 + 8y + y2 = 6 + 7y + y2 + (30 + 6y + (5y + y2))

Combine like terms: 6y + 5y = 11y
15 + 8y + y2 = 6 + 7y + y2 + (30 + 11y + y2)

Reorder the terms:
15 + 8y + y2 = 6 + 30 + 7y + 11y + y2 + y2

Combine like terms: 6 + 30 = 36
15 + 8y + y2 = 36 + 7y + 11y + y2 + y2

Combine like terms: 7y + 11y = 18y
15 + 8y + y2 = 36 + 18y + y2 + y2

Combine like terms: y2 + y2 = 2y2
15 + 8y + y2 = 36 + 18y + 2y2

Solving
15 + 8y + y2 = 36 + 18y + 2y2

Solving for variable 'y'.

Reorder the terms:
15 + -36 + 8y + -18y + y2 + -2y2 = 36 + 18y + 2y2 + -36 + -18y + -2y2

Combine like terms: 15 + -36 = -21
-21 + 8y + -18y + y2 + -2y2 = 36 + 18y + 2y2 + -36 + -18y + -2y2

Combine like terms: 8y + -18y = -10y
-21 + -10y + y2 + -2y2 = 36 + 18y + 2y2 + -36 + -18y + -2y2

Combine like terms: y2 + -2y2 = -1y2
-21 + -10y + -1y2 = 36 + 18y + 2y2 + -36 + -18y + -2y2

Reorder the terms:
-21 + -10y + -1y2 = 36 + -36 + 18y + -18y + 2y2 + -2y2

Combine like terms: 36 + -36 = 0
-21 + -10y + -1y2 = 0 + 18y + -18y + 2y2 + -2y2
-21 + -10y + -1y2 = 18y + -18y + 2y2 + -2y2

Combine like terms: 18y + -18y = 0
-21 + -10y + -1y2 = 0 + 2y2 + -2y2
-21 + -10y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
-21 + -10y + -1y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(21 + 10y + y2) = 0

Factor a trinomial.
-1((7 + y)(3 + y)) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(7 + y)' equal to zero and attempt to solve:

Simplifying
7 + y = 0

Solving
7 + y = 0

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

Add '-7' to each side of the equation.
7 + -7 + y = 0 + -7

Combine like terms: 7 + -7 = 0
0 + y = 0 + -7
y = 0 + -7

Combine like terms: 0 + -7 = -7
y = -7

Simplifying
y = -7
Subproblem 2
Set the factor '(3 + y)' equal to zero and attempt to solve:

Simplifying
3 + y = 0

Solving
3 + y = 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 + y = 0 + -3

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

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

Simplifying
y = -3
Solution
y = {-7, -3}