-3(n+7)(n+7)=-27

Solution for -3(n+7)(n+7)=-27 equation:

Simplifying
-3(n + 7)(n + 7) = -27

Reorder the terms:
-3(7 + n)(n + 7) = -27

Reorder the terms:
-3(7 + n)(7 + n) = -27

Multiply (7 + n) * (7 + n)
-3(7(7 + n) + n(7 + n)) = -27
-3((7 * 7 + n * 7) + n(7 + n)) = -27
-3((49 + 7n) + n(7 + n)) = -27
-3(49 + 7n + (7 * n + n * n)) = -27
-3(49 + 7n + (7n + n2)) = -27

Combine like terms: 7n + 7n = 14n
-3(49 + 14n + n2) = -27
(49 * -3 + 14n * -3 + n2 * -3) = -27
(-147 + -42n + -3n2) = -27

Solving
-147 + -42n + -3n2 = -27

Solving for variable 'n'.

Reorder the terms:
-147 + 27 + -42n + -3n2 = -27 + 27

Combine like terms: -147 + 27 = -120
-120 + -42n + -3n2 = -27 + 27

Combine like terms: -27 + 27 = 0
-120 + -42n + -3n2 = 0

Factor out the Greatest Common Factor (GCF), '-3'.
-3(40 + 14n + n2) = 0

Factor a trinomial.
-3((10 + n)(4 + n)) = 0

Ignore the factor -3.
Subproblem 1
Set the factor '(10 + n)' equal to zero and attempt to solve:

Simplifying
10 + n = 0

Solving
10 + n = 0

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

Add '-10' to each side of the equation.
10 + -10 + n = 0 + -10

Combine like terms: 10 + -10 = 0
0 + n = 0 + -10
n = 0 + -10

Combine like terms: 0 + -10 = -10
n = -10

Simplifying
n = -10
Subproblem 2
Set the factor '(4 + n)' equal to zero and attempt to solve:

Simplifying
4 + n = 0

Solving
4 + n = 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 + n = 0 + -4

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

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

Simplifying
n = -4
Solution
n = {-10, -4}