221=(n+1)(n+5)

Solution for 221=(n+1)(n+5) equation:

Simplifying
221 = (n + 1)(n + 5)

Reorder the terms:
221 = (1 + n)(n + 5)

Reorder the terms:
221 = (1 + n)(5 + n)

Multiply (1 + n) * (5 + n)
221 = (1(5 + n) + n(5 + n))
221 = ((5 * 1 + n * 1) + n(5 + n))
221 = ((5 + 1n) + n(5 + n))
221 = (5 + 1n + (5 * n + n * n))
221 = (5 + 1n + (5n + n2))

Combine like terms: 1n + 5n = 6n
221 = (5 + 6n + n2)

Solving
221 = 5 + 6n + n2

Solving for variable 'n'.

Combine like terms: 221 + -5 = 216
216 + -6n + -1n2 = 5 + 6n + n2 + -5 + -6n + -1n2

Reorder the terms:
216 + -6n + -1n2 = 5 + -5 + 6n + -6n + n2 + -1n2

Combine like terms: 5 + -5 = 0
216 + -6n + -1n2 = 0 + 6n + -6n + n2 + -1n2
216 + -6n + -1n2 = 6n + -6n + n2 + -1n2

Combine like terms: 6n + -6n = 0
216 + -6n + -1n2 = 0 + n2 + -1n2
216 + -6n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
216 + -6n + -1n2 = 0

Factor a trinomial.
(12 + -1n)(18 + n) = 0

Subproblem 1
Set the factor '(12 + -1n)' equal to zero and attempt to solve:

Simplifying
12 + -1n = 0

Solving
12 + -1n = 0

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

Add '-12' to each side of the equation.
12 + -12 + -1n = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -1n = 0 + -12
-1n = 0 + -12

Combine like terms: 0 + -12 = -12
-1n = -12

Divide each side by '-1'.
n = 12

Simplifying
n = 12
Subproblem 2
Set the factor '(18 + n)' equal to zero and attempt to solve:

Simplifying
18 + n = 0

Solving
18 + n = 0

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

Add '-18' to each side of the equation.
18 + -18 + n = 0 + -18

Combine like terms: 18 + -18 = 0
0 + n = 0 + -18
n = 0 + -18

Combine like terms: 0 + -18 = -18
n = -18

Simplifying
n = -18
Solution
n = {12, -18}