n(n+1)(2n+1)=0

Solution for n(n+1)(2n+1)=0 equation:

Simplifying
n(n + 1)(2n + 1) = 0

Reorder the terms:
n(1 + n)(2n + 1) = 0

Reorder the terms:
n(1 + n)(1 + 2n) = 0

Multiply (1 + n) * (1 + 2n)
n(1(1 + 2n) + n(1 + 2n)) = 0
n((1 * 1 + 2n * 1) + n(1 + 2n)) = 0
n((1 + 2n) + n(1 + 2n)) = 0
n(1 + 2n + (1 * n + 2n * n)) = 0
n(1 + 2n + (1n + 2n2)) = 0

Combine like terms: 2n + 1n = 3n
n(1 + 3n + 2n2) = 0
(1 * n + 3n * n + 2n2 * n) = 0
(1n + 3n2 + 2n3) = 0

Solving
1n + 3n2 + 2n3 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), 'n'.
n(1 + 3n + 2n2) = 0

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

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 '(1 + n)' equal to zero and attempt to solve:

Simplifying
1 + n = 0

Solving
1 + n = 0

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

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

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

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

Simplifying
n = -1
Subproblem 3
Set the factor '(1 + 2n)' equal to zero and attempt to solve:

Simplifying
1 + 2n = 0

Solving
1 + 2n = 0

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

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

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

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

Divide each side by '2'.
n = -0.5

Simplifying
n = -0.5
Solution
n = {0, -1, -0.5}