-3n-8n+7=n^2+4n

Solution for -3n-8n+7=n^2+4n equation:

Simplifying
-3n + -8n + 7 = n2 + 4n

Reorder the terms:
7 + -3n + -8n = n2 + 4n

Combine like terms: -3n + -8n = -11n
7 + -11n = n2 + 4n

Reorder the terms:
7 + -11n = 4n + n2

Solving
7 + -11n = 4n + n2

Solving for variable 'n'.

Combine like terms: -11n + -4n = -15n
7 + -15n + -1n2 = 4n + n2 + -4n + -1n2

Reorder the terms:
7 + -15n + -1n2 = 4n + -4n + n2 + -1n2

Combine like terms: 4n + -4n = 0
7 + -15n + -1n2 = 0 + n2 + -1n2
7 + -15n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
7 + -15n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-7 + 15n + n2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 15n + 7 + n2 = 0 + 7

Reorder the terms:
-7 + 7 + 15n + n2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 15n + n2 = 0 + 7
15n + n2 = 0 + 7

Combine like terms: 0 + 7 = 7
15n + n2 = 7

The n term is 15n.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15n + 56.25 + n2 = 7 + 56.25

Reorder the terms:
56.25 + 15n + n2 = 7 + 56.25

Combine like terms: 7 + 56.25 = 63.25
56.25 + 15n + n2 = 63.25

Factor a perfect square on the left side:
(n + 7.5)(n + 7.5) = 63.25

Calculate the square root of the right side: 7.95298686

Break this problem into two subproblems by setting 
(n + 7.5) equal to 7.95298686 and -7.95298686.

Subproblem 1
n + 7.5 = 7.95298686

Simplifying
n + 7.5 = 7.95298686

Reorder the terms:
7.5 + n = 7.95298686

Solving
7.5 + n = 7.95298686

Solving for variable 'n'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + n = 7.95298686 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + n = 7.95298686 + -7.5
n = 7.95298686 + -7.5

Combine like terms: 7.95298686 + -7.5 = 0.45298686
n = 0.45298686

Simplifying
n = 0.45298686

Subproblem 2
n + 7.5 = -7.95298686

Simplifying
n + 7.5 = -7.95298686

Reorder the terms:
7.5 + n = -7.95298686

Solving
7.5 + n = -7.95298686

Solving for variable 'n'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + n = -7.95298686 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + n = -7.95298686 + -7.5
n = -7.95298686 + -7.5

Combine like terms: -7.95298686 + -7.5 = -15.45298686
n = -15.45298686

Simplifying
n = -15.45298686

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.45298686, -15.45298686}