-2n(4n+5)=6(n-3)-8

Solution for -2n(4n+5)=6(n-3)-8 equation:

Simplifying
-2n(4n + 5) = 6(n + -3) + -8

Reorder the terms:
-2n(5 + 4n) = 6(n + -3) + -8
(5 * -2n + 4n * -2n) = 6(n + -3) + -8
(-10n + -8n2) = 6(n + -3) + -8

Reorder the terms:
-10n + -8n2 = 6(-3 + n) + -8
-10n + -8n2 = (-3 * 6 + n * 6) + -8
-10n + -8n2 = (-18 + 6n) + -8

Reorder the terms:
-10n + -8n2 = -18 + -8 + 6n

Combine like terms: -18 + -8 = -26
-10n + -8n2 = -26 + 6n

Solving
-10n + -8n2 = -26 + 6n

Solving for variable 'n'.

Reorder the terms:
26 + -10n + -6n + -8n2 = -26 + 6n + 26 + -6n

Combine like terms: -10n + -6n = -16n
26 + -16n + -8n2 = -26 + 6n + 26 + -6n

Reorder the terms:
26 + -16n + -8n2 = -26 + 26 + 6n + -6n

Combine like terms: -26 + 26 = 0
26 + -16n + -8n2 = 0 + 6n + -6n
26 + -16n + -8n2 = 6n + -6n

Combine like terms: 6n + -6n = 0
26 + -16n + -8n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(13 + -8n + -4n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(13 + -8n + -4n2)' equal to zero and attempt to solve:

Simplifying
13 + -8n + -4n2 = 0

Solving
13 + -8n + -4n2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-3.25 + 2n + n2 = 0

Move the constant term to the right:

Add '3.25' to each side of the equation.
-3.25 + 2n + 3.25 + n2 = 0 + 3.25

Reorder the terms:
-3.25 + 3.25 + 2n + n2 = 0 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + 2n + n2 = 0 + 3.25
2n + n2 = 0 + 3.25

Combine like terms: 0 + 3.25 = 3.25
2n + n2 = 3.25

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 3.25 + 1

Reorder the terms:
1 + 2n + n2 = 3.25 + 1

Combine like terms: 3.25 + 1 = 4.25
1 + 2n + n2 = 4.25

Factor a perfect square on the left side:
(n + 1)(n + 1) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(n + 1) equal to 2.061552813 and -2.061552813.

Subproblem 1
n + 1 = 2.061552813

Simplifying
n + 1 = 2.061552813

Reorder the terms:
1 + n = 2.061552813

Solving
1 + n = 2.061552813

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = 2.061552813 + -1

Combine like terms: 1 + -1 = 0
0 + n = 2.061552813 + -1
n = 2.061552813 + -1

Combine like terms: 2.061552813 + -1 = 1.061552813
n = 1.061552813

Simplifying
n = 1.061552813

Subproblem 2
n + 1 = -2.061552813

Simplifying
n + 1 = -2.061552813

Reorder the terms:
1 + n = -2.061552813

Solving
1 + n = -2.061552813

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = -2.061552813 + -1

Combine like terms: 1 + -1 = 0
0 + n = -2.061552813 + -1
n = -2.061552813 + -1

Combine like terms: -2.061552813 + -1 = -3.061552813
n = -3.061552813

Simplifying
n = -3.061552813

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.061552813, -3.061552813}
Solution
n = {1.061552813, -3.061552813}