(n+2)(n+4)-12n=0

Solution for (n+2)(n+4)-12n=0 equation:

Simplifying
(n + 2)(n + 4) + -12n = 0

Reorder the terms:
(2 + n)(n + 4) + -12n = 0

Reorder the terms:
(2 + n)(4 + n) + -12n = 0

Multiply (2 + n) * (4 + n)
(2(4 + n) + n(4 + n)) + -12n = 0
((4 * 2 + n * 2) + n(4 + n)) + -12n = 0
((8 + 2n) + n(4 + n)) + -12n = 0
(8 + 2n + (4 * n + n * n)) + -12n = 0
(8 + 2n + (4n + n2)) + -12n = 0

Combine like terms: 2n + 4n = 6n
(8 + 6n + n2) + -12n = 0

Reorder the terms:
8 + 6n + -12n + n2 = 0

Combine like terms: 6n + -12n = -6n
8 + -6n + n2 = 0

Solving
8 + -6n + n2 = 0

Solving for variable 'n'.

Factor a trinomial.
(2 + -1n)(4 + -1n) = 0

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

Simplifying
2 + -1n = 0

Solving
2 + -1n = 0

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

Add '-2' to each side of the equation.
2 + -2 + -1n = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -1n = 0 + -2
-1n = 0 + -2

Combine like terms: 0 + -2 = -2
-1n = -2

Divide each side by '-1'.
n = 2

Simplifying
n = 2
Subproblem 2
Set the factor '(4 + -1n)' equal to zero and attempt to solve:

Simplifying
4 + -1n = 0

Solving
4 + -1n = 0

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

Add '-4' to each side of the equation.
4 + -4 + -1n = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -1n = 0 + -4
-1n = 0 + -4

Combine like terms: 0 + -4 = -4
-1n = -4

Divide each side by '-1'.
n = 4

Simplifying
n = 4
Solution
n = {2, 4}