n^2+3n(n+6)=0

Solution for n^2+3n(n+6)=0 equation:

Simplifying
n2 + 3n(n + 6) = 0

Reorder the terms:
n2 + 3n(6 + n) = 0
n2 + (6 * 3n + n * 3n) = 0
n2 + (18n + 3n2) = 0

Reorder the terms:
18n + n2 + 3n2 = 0

Combine like terms: n2 + 3n2 = 4n2
18n + 4n2 = 0

Solving
18n + 4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2n'.
2n(9 + 2n) = 0

Ignore the factor 2.
Subproblem 1
Set the factor 'n' equal to zero and attempt to solve:

Simplifying
n = 0

Solving
n = 0

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

Simplifying
n = 0
Subproblem 2
Set the factor '(9 + 2n)' equal to zero and attempt to solve:

Simplifying
9 + 2n = 0

Solving
9 + 2n = 0

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

Add '-9' to each side of the equation.
9 + -9 + 2n = 0 + -9

Combine like terms: 9 + -9 = 0
0 + 2n = 0 + -9
2n = 0 + -9

Combine like terms: 0 + -9 = -9
2n = -9

Divide each side by '2'.
n = -4.5

Simplifying
n = -4.5
Solution
n = {0, -4.5}