k(k+1)(k+2)-3k(k+1)=0

Solution for k(k+1)(k+2)-3k(k+1)=0 equation:

Simplifying
k(k + 1)(k + 2) + -3k(k + 1) = 0

Reorder the terms:
k(1 + k)(k + 2) + -3k(k + 1) = 0

Reorder the terms:
k(1 + k)(2 + k) + -3k(k + 1) = 0

Multiply (1 + k) * (2 + k)
k(1(2 + k) + k(2 + k)) + -3k(k + 1) = 0
k((2 * 1 + k * 1) + k(2 + k)) + -3k(k + 1) = 0
k((2 + 1k) + k(2 + k)) + -3k(k + 1) = 0
k(2 + 1k + (2 * k + k * k)) + -3k(k + 1) = 0
k(2 + 1k + (2k + k2)) + -3k(k + 1) = 0

Combine like terms: 1k + 2k = 3k
k(2 + 3k + k2) + -3k(k + 1) = 0
(2 * k + 3k * k + k2 * k) + -3k(k + 1) = 0
(2k + 3k2 + k3) + -3k(k + 1) = 0

Reorder the terms:
2k + 3k2 + k3 + -3k(1 + k) = 0
2k + 3k2 + k3 + (1 * -3k + k * -3k) = 0
2k + 3k2 + k3 + (-3k + -3k2) = 0

Reorder the terms:
2k + -3k + 3k2 + -3k2 + k3 = 0

Combine like terms: 2k + -3k = -1k
-1k + 3k2 + -3k2 + k3 = 0

Combine like terms: 3k2 + -3k2 = 0
-1k + 0 + k3 = 0
-1k + k3 = 0

Solving
-1k + k3 = 0

Solving for variable 'k'.

Factor out the Greatest Common Factor (GCF), 'k'.
k(-1 + k2) = 0

Factor a difference between two squares.
k((1 + k)(-1 + k)) = 0

Subproblem 1
Set the factor 'k' equal to zero and attempt to solve:

Simplifying
k = 0

Solving
k = 0

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

Simplifying
k = 0
Subproblem 2
Set the factor '(1 + k)' equal to zero and attempt to solve:

Simplifying
1 + k = 0

Solving
1 + k = 0

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

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

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

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

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

Simplifying
-1 + k = 0

Solving
-1 + k = 0

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

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

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

Combine like terms: 0 + 1 = 1
k = 1

Simplifying
k = 1
Solution
k = {0, -1, 1}