2n(n+4)+(n-2)-15=3n(-n+1)+2n+6

Solution for 2n(n+4)+(n-2)-15=3n(-n+1)+2n+6 equation:

Simplifying
2n(n + 4) + (n + -2) + -15 = 3n(-1n + 1) + 2n + 6

Reorder the terms:
2n(4 + n) + (n + -2) + -15 = 3n(-1n + 1) + 2n + 6
(4 * 2n + n * 2n) + (n + -2) + -15 = 3n(-1n + 1) + 2n + 6
(8n + 2n2) + (n + -2) + -15 = 3n(-1n + 1) + 2n + 6

Reorder the terms:
8n + 2n2 + (-2 + n) + -15 = 3n(-1n + 1) + 2n + 6

Remove parenthesis around (-2 + n)
8n + 2n2 + -2 + n + -15 = 3n(-1n + 1) + 2n + 6

Reorder the terms:
-2 + -15 + 8n + n + 2n2 = 3n(-1n + 1) + 2n + 6

Combine like terms: -2 + -15 = -17
-17 + 8n + n + 2n2 = 3n(-1n + 1) + 2n + 6

Combine like terms: 8n + n = 9n
-17 + 9n + 2n2 = 3n(-1n + 1) + 2n + 6

Reorder the terms:
-17 + 9n + 2n2 = 3n(1 + -1n) + 2n + 6
-17 + 9n + 2n2 = (1 * 3n + -1n * 3n) + 2n + 6
-17 + 9n + 2n2 = (3n + -3n2) + 2n + 6

Reorder the terms:
-17 + 9n + 2n2 = 6 + 3n + 2n + -3n2

Combine like terms: 3n + 2n = 5n
-17 + 9n + 2n2 = 6 + 5n + -3n2

Solving
-17 + 9n + 2n2 = 6 + 5n + -3n2

Solving for variable 'n'.

Reorder the terms:
-17 + -6 + 9n + -5n + 2n2 + 3n2 = 6 + 5n + -3n2 + -6 + -5n + 3n2

Combine like terms: -17 + -6 = -23
-23 + 9n + -5n + 2n2 + 3n2 = 6 + 5n + -3n2 + -6 + -5n + 3n2

Combine like terms: 9n + -5n = 4n
-23 + 4n + 2n2 + 3n2 = 6 + 5n + -3n2 + -6 + -5n + 3n2

Combine like terms: 2n2 + 3n2 = 5n2
-23 + 4n + 5n2 = 6 + 5n + -3n2 + -6 + -5n + 3n2

Reorder the terms:
-23 + 4n + 5n2 = 6 + -6 + 5n + -5n + -3n2 + 3n2

Combine like terms: 6 + -6 = 0
-23 + 4n + 5n2 = 0 + 5n + -5n + -3n2 + 3n2
-23 + 4n + 5n2 = 5n + -5n + -3n2 + 3n2

Combine like terms: 5n + -5n = 0
-23 + 4n + 5n2 = 0 + -3n2 + 3n2
-23 + 4n + 5n2 = -3n2 + 3n2

Combine like terms: -3n2 + 3n2 = 0
-23 + 4n + 5n2 = 0

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

Divide each side by '5'.
-4.6 + 0.8n + n2 = 0

Move the constant term to the right:

Add '4.6' to each side of the equation.
-4.6 + 0.8n + 4.6 + n2 = 0 + 4.6

Reorder the terms:
-4.6 + 4.6 + 0.8n + n2 = 0 + 4.6

Combine like terms: -4.6 + 4.6 = 0.0
0.0 + 0.8n + n2 = 0 + 4.6
0.8n + n2 = 0 + 4.6

Combine like terms: 0 + 4.6 = 4.6
0.8n + n2 = 4.6

The n term is 0.8n.  Take half its coefficient (0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
0.8n + 0.16 + n2 = 4.6 + 0.16

Reorder the terms:
0.16 + 0.8n + n2 = 4.6 + 0.16

Combine like terms: 4.6 + 0.16 = 4.76
0.16 + 0.8n + n2 = 4.76

Factor a perfect square on the left side:
(n + 0.4)(n + 0.4) = 4.76

Calculate the square root of the right side: 2.181742423

Break this problem into two subproblems by setting 
(n + 0.4) equal to 2.181742423 and -2.181742423.

Subproblem 1
n + 0.4 = 2.181742423

Simplifying
n + 0.4 = 2.181742423

Reorder the terms:
0.4 + n = 2.181742423

Solving
0.4 + n = 2.181742423

Solving for variable 'n'.

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

Add '-0.4' to each side of the equation.
0.4 + -0.4 + n = 2.181742423 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + n = 2.181742423 + -0.4
n = 2.181742423 + -0.4

Combine like terms: 2.181742423 + -0.4 = 1.781742423
n = 1.781742423

Simplifying
n = 1.781742423

Subproblem 2
n + 0.4 = -2.181742423

Simplifying
n + 0.4 = -2.181742423

Reorder the terms:
0.4 + n = -2.181742423

Solving
0.4 + n = -2.181742423

Solving for variable 'n'.

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

Add '-0.4' to each side of the equation.
0.4 + -0.4 + n = -2.181742423 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + n = -2.181742423 + -0.4
n = -2.181742423 + -0.4

Combine like terms: -2.181742423 + -0.4 = -2.581742423
n = -2.581742423

Simplifying
n = -2.581742423

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.781742423, -2.581742423}