(5y+3)(5y+4)=6(2y+6)

Solution for (5y+3)(5y+4)=6(2y+6) equation:

Simplifying
(5y + 3)(5y + 4) = 6(2y + 6)

Reorder the terms:
(3 + 5y)(5y + 4) = 6(2y + 6)

Reorder the terms:
(3 + 5y)(4 + 5y) = 6(2y + 6)

Multiply (3 + 5y) * (4 + 5y)
(3(4 + 5y) + 5y * (4 + 5y)) = 6(2y + 6)
((4 * 3 + 5y * 3) + 5y * (4 + 5y)) = 6(2y + 6)
((12 + 15y) + 5y * (4 + 5y)) = 6(2y + 6)
(12 + 15y + (4 * 5y + 5y * 5y)) = 6(2y + 6)
(12 + 15y + (20y + 25y2)) = 6(2y + 6)

Combine like terms: 15y + 20y = 35y
(12 + 35y + 25y2) = 6(2y + 6)

Reorder the terms:
12 + 35y + 25y2 = 6(6 + 2y)
12 + 35y + 25y2 = (6 * 6 + 2y * 6)
12 + 35y + 25y2 = (36 + 12y)

Solving
12 + 35y + 25y2 = 36 + 12y

Solving for variable 'y'.

Reorder the terms:
12 + -36 + 35y + -12y + 25y2 = 36 + 12y + -36 + -12y

Combine like terms: 12 + -36 = -24
-24 + 35y + -12y + 25y2 = 36 + 12y + -36 + -12y

Combine like terms: 35y + -12y = 23y
-24 + 23y + 25y2 = 36 + 12y + -36 + -12y

Reorder the terms:
-24 + 23y + 25y2 = 36 + -36 + 12y + -12y

Combine like terms: 36 + -36 = 0
-24 + 23y + 25y2 = 0 + 12y + -12y
-24 + 23y + 25y2 = 12y + -12y

Combine like terms: 12y + -12y = 0
-24 + 23y + 25y2 = 0

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

Divide each side by '25'.
-0.96 + 0.92y + y2 = 0

Move the constant term to the right:

Add '0.96' to each side of the equation.
-0.96 + 0.92y + 0.96 + y2 = 0 + 0.96

Reorder the terms:
-0.96 + 0.96 + 0.92y + y2 = 0 + 0.96

Combine like terms: -0.96 + 0.96 = 0.00
0.00 + 0.92y + y2 = 0 + 0.96
0.92y + y2 = 0 + 0.96

Combine like terms: 0 + 0.96 = 0.96
0.92y + y2 = 0.96

The y term is 0.92y.  Take half its coefficient (0.46).
Square it (0.2116) and add it to both sides.

Add '0.2116' to each side of the equation.
0.92y + 0.2116 + y2 = 0.96 + 0.2116

Reorder the terms:
0.2116 + 0.92y + y2 = 0.96 + 0.2116

Combine like terms: 0.96 + 0.2116 = 1.1716
0.2116 + 0.92y + y2 = 1.1716

Factor a perfect square on the left side:
(y + 0.46)(y + 0.46) = 1.1716

Calculate the square root of the right side: 1.08240473

Break this problem into two subproblems by setting 
(y + 0.46) equal to 1.08240473 and -1.08240473.

Subproblem 1
y + 0.46 = 1.08240473

Simplifying
y + 0.46 = 1.08240473

Reorder the terms:
0.46 + y = 1.08240473

Solving
0.46 + y = 1.08240473

Solving for variable 'y'.

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

Add '-0.46' to each side of the equation.
0.46 + -0.46 + y = 1.08240473 + -0.46

Combine like terms: 0.46 + -0.46 = 0.00
0.00 + y = 1.08240473 + -0.46
y = 1.08240473 + -0.46

Combine like terms: 1.08240473 + -0.46 = 0.62240473
y = 0.62240473

Simplifying
y = 0.62240473

Subproblem 2
y + 0.46 = -1.08240473

Simplifying
y + 0.46 = -1.08240473

Reorder the terms:
0.46 + y = -1.08240473

Solving
0.46 + y = -1.08240473

Solving for variable 'y'.

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

Add '-0.46' to each side of the equation.
0.46 + -0.46 + y = -1.08240473 + -0.46

Combine like terms: 0.46 + -0.46 = 0.00
0.00 + y = -1.08240473 + -0.46
y = -1.08240473 + -0.46

Combine like terms: -1.08240473 + -0.46 = -1.54240473
y = -1.54240473

Simplifying
y = -1.54240473

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.62240473, -1.54240473}