(y-2)(y+1)+(y-5)(y-3)=14y+7

Solution for (y-2)(y+1)+(y-5)(y-3)=14y+7 equation:

Simplifying
(y + -2)(y + 1) + (y + -5)(y + -3) = 14y + 7

Reorder the terms:
(-2 + y)(y + 1) + (y + -5)(y + -3) = 14y + 7

Reorder the terms:
(-2 + y)(1 + y) + (y + -5)(y + -3) = 14y + 7

Multiply (-2 + y) * (1 + y)
(-2(1 + y) + y(1 + y)) + (y + -5)(y + -3) = 14y + 7
((1 * -2 + y * -2) + y(1 + y)) + (y + -5)(y + -3) = 14y + 7
((-2 + -2y) + y(1 + y)) + (y + -5)(y + -3) = 14y + 7
(-2 + -2y + (1 * y + y * y)) + (y + -5)(y + -3) = 14y + 7
(-2 + -2y + (1y + y2)) + (y + -5)(y + -3) = 14y + 7

Combine like terms: -2y + 1y = -1y
(-2 + -1y + y2) + (y + -5)(y + -3) = 14y + 7

Reorder the terms:
-2 + -1y + y2 + (-5 + y)(y + -3) = 14y + 7

Reorder the terms:
-2 + -1y + y2 + (-5 + y)(-3 + y) = 14y + 7

Multiply (-5 + y) * (-3 + y)
-2 + -1y + y2 + (-5(-3 + y) + y(-3 + y)) = 14y + 7
-2 + -1y + y2 + ((-3 * -5 + y * -5) + y(-3 + y)) = 14y + 7
-2 + -1y + y2 + ((15 + -5y) + y(-3 + y)) = 14y + 7
-2 + -1y + y2 + (15 + -5y + (-3 * y + y * y)) = 14y + 7
-2 + -1y + y2 + (15 + -5y + (-3y + y2)) = 14y + 7

Combine like terms: -5y + -3y = -8y
-2 + -1y + y2 + (15 + -8y + y2) = 14y + 7

Reorder the terms:
-2 + 15 + -1y + -8y + y2 + y2 = 14y + 7

Combine like terms: -2 + 15 = 13
13 + -1y + -8y + y2 + y2 = 14y + 7

Combine like terms: -1y + -8y = -9y
13 + -9y + y2 + y2 = 14y + 7

Combine like terms: y2 + y2 = 2y2
13 + -9y + 2y2 = 14y + 7

Reorder the terms:
13 + -9y + 2y2 = 7 + 14y

Solving
13 + -9y + 2y2 = 7 + 14y

Solving for variable 'y'.

Reorder the terms:
13 + -7 + -9y + -14y + 2y2 = 7 + 14y + -7 + -14y

Combine like terms: 13 + -7 = 6
6 + -9y + -14y + 2y2 = 7 + 14y + -7 + -14y

Combine like terms: -9y + -14y = -23y
6 + -23y + 2y2 = 7 + 14y + -7 + -14y

Reorder the terms:
6 + -23y + 2y2 = 7 + -7 + 14y + -14y

Combine like terms: 7 + -7 = 0
6 + -23y + 2y2 = 0 + 14y + -14y
6 + -23y + 2y2 = 14y + -14y

Combine like terms: 14y + -14y = 0
6 + -23y + 2y2 = 0

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

Divide each side by '2'.
3 + -11.5y + y2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -11.5y + -3 + y2 = 0 + -3

Reorder the terms:
3 + -3 + -11.5y + y2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -11.5y + y2 = 0 + -3
-11.5y + y2 = 0 + -3

Combine like terms: 0 + -3 = -3
-11.5y + y2 = -3

The y term is -11.5y.  Take half its coefficient (-5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
-11.5y + 33.0625 + y2 = -3 + 33.0625

Reorder the terms:
33.0625 + -11.5y + y2 = -3 + 33.0625

Combine like terms: -3 + 33.0625 = 30.0625
33.0625 + -11.5y + y2 = 30.0625

Factor a perfect square on the left side:
(y + -5.75)(y + -5.75) = 30.0625

Calculate the square root of the right side: 5.48292805

Break this problem into two subproblems by setting 
(y + -5.75) equal to 5.48292805 and -5.48292805.

Subproblem 1
y + -5.75 = 5.48292805

Simplifying
y + -5.75 = 5.48292805

Reorder the terms:
-5.75 + y = 5.48292805

Solving
-5.75 + y = 5.48292805

Solving for variable 'y'.

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

Add '5.75' to each side of the equation.
-5.75 + 5.75 + y = 5.48292805 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + y = 5.48292805 + 5.75
y = 5.48292805 + 5.75

Combine like terms: 5.48292805 + 5.75 = 11.23292805
y = 11.23292805

Simplifying
y = 11.23292805

Subproblem 2
y + -5.75 = -5.48292805

Simplifying
y + -5.75 = -5.48292805

Reorder the terms:
-5.75 + y = -5.48292805

Solving
-5.75 + y = -5.48292805

Solving for variable 'y'.

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

Add '5.75' to each side of the equation.
-5.75 + 5.75 + y = -5.48292805 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + y = -5.48292805 + 5.75
y = -5.48292805 + 5.75

Combine like terms: -5.48292805 + 5.75 = 0.26707195
y = 0.26707195

Simplifying
y = 0.26707195

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {11.23292805, 0.26707195}