-3n(n-3)(-4n-6)=-8n

Solution for -3n(n-3)(-4n-6)=-8n equation:

Simplifying
-3n(n + -3)(-4n + -6) = -8n

Reorder the terms:
-3n(-3 + n)(-4n + -6) = -8n

Reorder the terms:
-3n(-3 + n)(-6 + -4n) = -8n

Multiply (-3 + n) * (-6 + -4n)
-3n(-3(-6 + -4n) + n(-6 + -4n)) = -8n
-3n((-6 * -3 + -4n * -3) + n(-6 + -4n)) = -8n
-3n((18 + 12n) + n(-6 + -4n)) = -8n
-3n(18 + 12n + (-6 * n + -4n * n)) = -8n
-3n(18 + 12n + (-6n + -4n2)) = -8n

Combine like terms: 12n + -6n = 6n
-3n(18 + 6n + -4n2) = -8n
(18 * -3n + 6n * -3n + -4n2 * -3n) = -8n
(-54n + -18n2 + 12n3) = -8n

Solving
-54n + -18n2 + 12n3 = -8n

Solving for variable 'n'.

Reorder the terms:
-54n + 8n + -18n2 + 12n3 = -8n + 8n

Combine like terms: -54n + 8n = -46n
-46n + -18n2 + 12n3 = -8n + 8n

Combine like terms: -8n + 8n = 0
-46n + -18n2 + 12n3 = 0

Factor out the Greatest Common Factor (GCF), '2n'.
2n(-23 + -9n + 6n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor 'n' equal to zero and attempt to solve:

Simplifying
n = 0

Solving
n = 0

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

Simplifying
n = 0
Subproblem 2
Set the factor '(-23 + -9n + 6n2)' equal to zero and attempt to solve:

Simplifying
-23 + -9n + 6n2 = 0

Solving
-23 + -9n + 6n2 = 0

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

Divide each side by '6'.
-3.833333333 + -1.5n + n2 = 0

Move the constant term to the right:

Add '3.833333333' to each side of the equation.
-3.833333333 + -1.5n + 3.833333333 + n2 = 0 + 3.833333333

Reorder the terms:
-3.833333333 + 3.833333333 + -1.5n + n2 = 0 + 3.833333333

Combine like terms: -3.833333333 + 3.833333333 = 0.000000000
0.000000000 + -1.5n + n2 = 0 + 3.833333333
-1.5n + n2 = 0 + 3.833333333

Combine like terms: 0 + 3.833333333 = 3.833333333
-1.5n + n2 = 3.833333333

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = 3.833333333 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 3.833333333 + 0.5625

Combine like terms: 3.833333333 + 0.5625 = 4.395833333
0.5625 + -1.5n + n2 = 4.395833333

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 4.395833333

Calculate the square root of the right side: 2.096624271

Break this problem into two subproblems by setting 
(n + -0.75) equal to 2.096624271 and -2.096624271.

Subproblem 1
n + -0.75 = 2.096624271

Simplifying
n + -0.75 = 2.096624271

Reorder the terms:
-0.75 + n = 2.096624271

Solving
-0.75 + n = 2.096624271

Solving for variable 'n'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + n = 2.096624271 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + n = 2.096624271 + 0.75
n = 2.096624271 + 0.75

Combine like terms: 2.096624271 + 0.75 = 2.846624271
n = 2.846624271

Simplifying
n = 2.846624271

Subproblem 2
n + -0.75 = -2.096624271

Simplifying
n + -0.75 = -2.096624271

Reorder the terms:
-0.75 + n = -2.096624271

Solving
-0.75 + n = -2.096624271

Solving for variable 'n'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + n = -2.096624271 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + n = -2.096624271 + 0.75
n = -2.096624271 + 0.75

Combine like terms: -2.096624271 + 0.75 = -1.346624271
n = -1.346624271

Simplifying
n = -1.346624271

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.846624271, -1.346624271}
Solution
n = {0, 2.846624271, -1.346624271}