n(n+1)+(n+2)=258

Solution for n(n+1)+(n+2)=258 equation:

Simplifying
n(n + 1) + (n + 2) = 258

Reorder the terms:
n(1 + n) + (n + 2) = 258
(1 * n + n * n) + (n + 2) = 258
(1n + n2) + (n + 2) = 258

Reorder the terms:
1n + n2 + (2 + n) = 258

Remove parenthesis around (2 + n)
1n + n2 + 2 + n = 258

Reorder the terms:
2 + 1n + n + n2 = 258

Combine like terms: 1n + n = 2n
2 + 2n + n2 = 258

Solving
2 + 2n + n2 = 258

Solving for variable 'n'.

Reorder the terms:
2 + -258 + 2n + n2 = 258 + -258

Combine like terms: 2 + -258 = -256
-256 + 2n + n2 = 258 + -258

Combine like terms: 258 + -258 = 0
-256 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '256' to each side of the equation.
-256 + 2n + 256 + n2 = 0 + 256

Reorder the terms:
-256 + 256 + 2n + n2 = 0 + 256

Combine like terms: -256 + 256 = 0
0 + 2n + n2 = 0 + 256
2n + n2 = 0 + 256

Combine like terms: 0 + 256 = 256
2n + n2 = 256

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 = 256 + 1

Reorder the terms:
1 + 2n + n2 = 256 + 1

Combine like terms: 256 + 1 = 257
1 + 2n + n2 = 257

Factor a perfect square on the left side:
(n + 1)(n + 1) = 257

Calculate the square root of the right side: 16.031219542

Break this problem into two subproblems by setting 
(n + 1) equal to 16.031219542 and -16.031219542.

Subproblem 1
n + 1 = 16.031219542

Simplifying
n + 1 = 16.031219542

Reorder the terms:
1 + n = 16.031219542

Solving
1 + n = 16.031219542

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 = 16.031219542 + -1

Combine like terms: 1 + -1 = 0
0 + n = 16.031219542 + -1
n = 16.031219542 + -1

Combine like terms: 16.031219542 + -1 = 15.031219542
n = 15.031219542

Simplifying
n = 15.031219542

Subproblem 2
n + 1 = -16.031219542

Simplifying
n + 1 = -16.031219542

Reorder the terms:
1 + n = -16.031219542

Solving
1 + n = -16.031219542

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 = -16.031219542 + -1

Combine like terms: 1 + -1 = 0
0 + n = -16.031219542 + -1
n = -16.031219542 + -1

Combine like terms: -16.031219542 + -1 = -17.031219542
n = -17.031219542

Simplifying
n = -17.031219542

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {15.031219542, -17.031219542}