(n+1)(n+3)=116

Solution for (n+1)(n+3)=116 equation:

Simplifying
(n + 1)(n + 3) = 116

Reorder the terms:
(1 + n)(n + 3) = 116

Reorder the terms:
(1 + n)(3 + n) = 116

Multiply (1 + n) * (3 + n)
(1(3 + n) + n(3 + n)) = 116
((3 * 1 + n * 1) + n(3 + n)) = 116
((3 + 1n) + n(3 + n)) = 116
(3 + 1n + (3 * n + n * n)) = 116
(3 + 1n + (3n + n2)) = 116

Combine like terms: 1n + 3n = 4n
(3 + 4n + n2) = 116

Solving
3 + 4n + n2 = 116

Solving for variable 'n'.

Reorder the terms:
3 + -116 + 4n + n2 = 116 + -116

Combine like terms: 3 + -116 = -113
-113 + 4n + n2 = 116 + -116

Combine like terms: 116 + -116 = 0
-113 + 4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '113' to each side of the equation.
-113 + 4n + 113 + n2 = 0 + 113

Reorder the terms:
-113 + 113 + 4n + n2 = 0 + 113

Combine like terms: -113 + 113 = 0
0 + 4n + n2 = 0 + 113
4n + n2 = 0 + 113

Combine like terms: 0 + 113 = 113
4n + n2 = 113

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = 113 + 4

Reorder the terms:
4 + 4n + n2 = 113 + 4

Combine like terms: 113 + 4 = 117
4 + 4n + n2 = 117

Factor a perfect square on the left side:
(n + 2)(n + 2) = 117

Calculate the square root of the right side: 10.816653826

Break this problem into two subproblems by setting 
(n + 2) equal to 10.816653826 and -10.816653826.

Subproblem 1
n + 2 = 10.816653826

Simplifying
n + 2 = 10.816653826

Reorder the terms:
2 + n = 10.816653826

Solving
2 + n = 10.816653826

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = 10.816653826 + -2

Combine like terms: 2 + -2 = 0
0 + n = 10.816653826 + -2
n = 10.816653826 + -2

Combine like terms: 10.816653826 + -2 = 8.816653826
n = 8.816653826

Simplifying
n = 8.816653826

Subproblem 2
n + 2 = -10.816653826

Simplifying
n + 2 = -10.816653826

Reorder the terms:
2 + n = -10.816653826

Solving
2 + n = -10.816653826

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = -10.816653826 + -2

Combine like terms: 2 + -2 = 0
0 + n = -10.816653826 + -2
n = -10.816653826 + -2

Combine like terms: -10.816653826 + -2 = -12.816653826
n = -12.816653826

Simplifying
n = -12.816653826

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {8.816653826, -12.816653826}