n(n+1)+(n+1+1)=27

Solution for n(n+1)+(n+1+1)=27 equation:

Simplifying
n(n + 1) + (n + 1 + 1) = 27

Reorder the terms:
n(1 + n) + (n + 1 + 1) = 27
(1 * n + n * n) + (n + 1 + 1) = 27
(1n + n2) + (n + 1 + 1) = 27

Reorder the terms:
1n + n2 + (1 + 1 + n) = 27

Combine like terms: 1 + 1 = 2
1n + n2 + (2 + n) = 27

Remove parenthesis around (2 + n)
1n + n2 + 2 + n = 27

Reorder the terms:
2 + 1n + n + n2 = 27

Combine like terms: 1n + n = 2n
2 + 2n + n2 = 27

Solving
2 + 2n + n2 = 27

Solving for variable 'n'.

Reorder the terms:
2 + -27 + 2n + n2 = 27 + -27

Combine like terms: 2 + -27 = -25
-25 + 2n + n2 = 27 + -27

Combine like terms: 27 + -27 = 0
-25 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 2n + 25 + n2 = 0 + 25

Reorder the terms:
-25 + 25 + 2n + n2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 2n + n2 = 0 + 25
2n + n2 = 0 + 25

Combine like terms: 0 + 25 = 25
2n + n2 = 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 = 25 + 1

Reorder the terms:
1 + 2n + n2 = 25 + 1

Combine like terms: 25 + 1 = 26
1 + 2n + n2 = 26

Factor a perfect square on the left side:
(n + 1)(n + 1) = 26

Calculate the square root of the right side: 5.099019514

Break this problem into two subproblems by setting 
(n + 1) equal to 5.099019514 and -5.099019514.

Subproblem 1
n + 1 = 5.099019514

Simplifying
n + 1 = 5.099019514

Reorder the terms:
1 + n = 5.099019514

Solving
1 + n = 5.099019514

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 = 5.099019514 + -1

Combine like terms: 1 + -1 = 0
0 + n = 5.099019514 + -1
n = 5.099019514 + -1

Combine like terms: 5.099019514 + -1 = 4.099019514
n = 4.099019514

Simplifying
n = 4.099019514

Subproblem 2
n + 1 = -5.099019514

Simplifying
n + 1 = -5.099019514

Reorder the terms:
1 + n = -5.099019514

Solving
1 + n = -5.099019514

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 = -5.099019514 + -1

Combine like terms: 1 + -1 = 0
0 + n = -5.099019514 + -1
n = -5.099019514 + -1

Combine like terms: -5.099019514 + -1 = -6.099019514
n = -6.099019514

Simplifying
n = -6.099019514

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.099019514, -6.099019514}