Solve the Cubic Equation: x³ = x² + 2x Step-by-Step

$x^3=x^2+2x$

To solve the problem $x^3 = x^2 + 2x$, follow these steps:

• Step 1: Re-arrange the equation to have all terms on one side:
  $x^3 - x^2 - 2x = 0$.
• Step 2: Factor out the greatest common factor (GCF), which is $x$:
  $x(x^2 - x - 2) = 0$.
• Step 3: Factor the quadratic expression $x^2 - x - 2$:
  The factors of $-2$ that add up to $-1$ are $-2$ and $1$. Thus, $x^2 - x - 2 = (x-2)(x+1)$.
• Step 4: Combine the factored terms:
  $x(x-2)(x+1) = 0$.

Each factor can be set to zero to find the solutions:

• $x = 0$.
• $x - 2 = 0$, so $x = 2$.
• $x + 1 = 0$, so $x = -1$.

The solutions to the equation are $x = 0, -1, 2$.

Therefore, the correct choice from the given options is:
$x = 0, -1, 2$.