n(n+2)=71+7(n+n+2)

Solution for n(n+2)=71+7(n+n+2) equation:

Simplifying
n(n + 2) = 71 + 7(n + n + 2)

Reorder the terms:
n(2 + n) = 71 + 7(n + n + 2)
(2 * n + n * n) = 71 + 7(n + n + 2)
(2n + n2) = 71 + 7(n + n + 2)

Reorder the terms:
2n + n2 = 71 + 7(2 + n + n)

Combine like terms: n + n = 2n
2n + n2 = 71 + 7(2 + 2n)
2n + n2 = 71 + (2 * 7 + 2n * 7)
2n + n2 = 71 + (14 + 14n)

Combine like terms: 71 + 14 = 85
2n + n2 = 85 + 14n

Solving
2n + n2 = 85 + 14n

Solving for variable 'n'.

Reorder the terms:
-85 + 2n + -14n + n2 = 85 + 14n + -85 + -14n

Combine like terms: 2n + -14n = -12n
-85 + -12n + n2 = 85 + 14n + -85 + -14n

Reorder the terms:
-85 + -12n + n2 = 85 + -85 + 14n + -14n

Combine like terms: 85 + -85 = 0
-85 + -12n + n2 = 0 + 14n + -14n
-85 + -12n + n2 = 14n + -14n

Combine like terms: 14n + -14n = 0
-85 + -12n + n2 = 0

Factor a trinomial.
(-5 + -1n)(17 + -1n) = 0

Subproblem 1
Set the factor '(-5 + -1n)' equal to zero and attempt to solve:

Simplifying
-5 + -1n = 0

Solving
-5 + -1n = 0

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

Add '5' to each side of the equation.
-5 + 5 + -1n = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1n = 0 + 5
-1n = 0 + 5

Combine like terms: 0 + 5 = 5
-1n = 5

Divide each side by '-1'.
n = -5

Simplifying
n = -5
Subproblem 2
Set the factor '(17 + -1n)' equal to zero and attempt to solve:

Simplifying
17 + -1n = 0

Solving
17 + -1n = 0

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

Add '-17' to each side of the equation.
17 + -17 + -1n = 0 + -17

Combine like terms: 17 + -17 = 0
0 + -1n = 0 + -17
-1n = 0 + -17

Combine like terms: 0 + -17 = -17
-1n = -17

Divide each side by '-1'.
n = 17

Simplifying
n = 17
Solution
n = {-5, 17}