4(y+3)(-2y)=7

Solution for 4(y+3)(-2y)=7 equation:

Simplifying
4(y + 3)(-2y) = 7

Reorder the terms:
4(3 + y)(-2y) = 7

Remove parenthesis around (-2y)
4(3 + y) * -2y = 7

Reorder the terms for easier multiplication:
4 * -2y(3 + y) = 7

Multiply 4 * -2
-8y(3 + y) = 7
(3 * -8y + y * -8y) = 7
(-24y + -8y2) = 7

Solving
-24y + -8y2 = 7

Solving for variable 'y'.

Reorder the terms:
-7 + -24y + -8y2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + -24y + -8y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(7 + 24y + 8y2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(7 + 24y + 8y2)' equal to zero and attempt to solve:

Simplifying
7 + 24y + 8y2 = 0

Solving
7 + 24y + 8y2 = 0

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

Divide each side by '8'.
0.875 + 3y + y2 = 0

Move the constant term to the right:

Add '-0.875' to each side of the equation.
0.875 + 3y + -0.875 + y2 = 0 + -0.875

Reorder the terms:
0.875 + -0.875 + 3y + y2 = 0 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + 3y + y2 = 0 + -0.875
3y + y2 = 0 + -0.875

Combine like terms: 0 + -0.875 = -0.875
3y + y2 = -0.875

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = -0.875 + 2.25

Reorder the terms:
2.25 + 3y + y2 = -0.875 + 2.25

Combine like terms: -0.875 + 2.25 = 1.375
2.25 + 3y + y2 = 1.375

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 1.375

Calculate the square root of the right side: 1.17260394

Break this problem into two subproblems by setting 
(y + 1.5) equal to 1.17260394 and -1.17260394.

Subproblem 1
y + 1.5 = 1.17260394

Simplifying
y + 1.5 = 1.17260394

Reorder the terms:
1.5 + y = 1.17260394

Solving
1.5 + y = 1.17260394

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 1.17260394 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 1.17260394 + -1.5
y = 1.17260394 + -1.5

Combine like terms: 1.17260394 + -1.5 = -0.32739606
y = -0.32739606

Simplifying
y = -0.32739606

Subproblem 2
y + 1.5 = -1.17260394

Simplifying
y + 1.5 = -1.17260394

Reorder the terms:
1.5 + y = -1.17260394

Solving
1.5 + y = -1.17260394

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -1.17260394 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -1.17260394 + -1.5
y = -1.17260394 + -1.5

Combine like terms: -1.17260394 + -1.5 = -2.67260394
y = -2.67260394

Simplifying
y = -2.67260394

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.32739606, -2.67260394}
Solution
y = {-0.32739606, -2.67260394}