n(n+2)=6(n+n+2)-1

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

Simplifying
n(n + 2) = 6(n + n + 2) + -1

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

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

Combine like terms: n + n = 2n
2n + n2 = 6(2 + 2n) + -1
2n + n2 = (2 * 6 + 2n * 6) + -1
2n + n2 = (12 + 12n) + -1

Reorder the terms:
2n + n2 = 12 + -1 + 12n

Combine like terms: 12 + -1 = 11
2n + n2 = 11 + 12n

Solving
2n + n2 = 11 + 12n

Solving for variable 'n'.

Reorder the terms:
-11 + 2n + -12n + n2 = 11 + 12n + -11 + -12n

Combine like terms: 2n + -12n = -10n
-11 + -10n + n2 = 11 + 12n + -11 + -12n

Reorder the terms:
-11 + -10n + n2 = 11 + -11 + 12n + -12n

Combine like terms: 11 + -11 = 0
-11 + -10n + n2 = 0 + 12n + -12n
-11 + -10n + n2 = 12n + -12n

Combine like terms: 12n + -12n = 0
-11 + -10n + n2 = 0

Factor a trinomial.
(-1 + -1n)(11 + -1n) = 0

Subproblem 1
Set the factor '(-1 + -1n)' equal to zero and attempt to solve:

Simplifying
-1 + -1n = 0

Solving
-1 + -1n = 0

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

Add '1' to each side of the equation.
-1 + 1 + -1n = 0 + 1

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

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

Divide each side by '-1'.
n = -1

Simplifying
n = -1
Subproblem 2
Set the factor '(11 + -1n)' equal to zero and attempt to solve:

Simplifying
11 + -1n = 0

Solving
11 + -1n = 0

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

Add '-11' to each side of the equation.
11 + -11 + -1n = 0 + -11

Combine like terms: 11 + -11 = 0
0 + -1n = 0 + -11
-1n = 0 + -11

Combine like terms: 0 + -11 = -11
-1n = -11

Divide each side by '-1'.
n = 11

Simplifying
n = 11
Solution
n = {-1, 11}