The given equation is:

$x^3 - 2x^2 - 3x = 0$

We are tasked to factor the equation first and then solve for $x$.

Step 1: Factor the equation

Start by factoring out the greatest common factor (GCF) of all terms, which is $x$: $x(x^2 - 2x - 3) = 0$

Next, factorize the quadratic expression $x^2 - 2x - 3$. Look for two numbers whose product is $-3$ and sum is $-2$. These numbers are $-3$ and $1$.

Thus: $x^2 - 2x - 3 = (x - 3)(x + 1)$

So the fully factored form of the equation is: $x(x - 3)(x + 1) = 0$

Step 2: Solve for $x$

Set each factor equal to zero:

1. $x = 0$
2. $x - 3 = 0 \implies x = 3$
3. $x + 1 = 0 \implies x = -1$

Final Solution:

The solutions to the equation are: $x = 0, \quad x = 3, \quad x = -1$

Would you like a verification or more details about this process?

Related Questions:

1. How do you verify the solutions of a polynomial equation?
2. What does it mean when a polynomial has multiple roots?
3. Can you explain the difference between factoring and solving?
4. How does the quadratic formula relate to solving quadratic expressions?
5. How would this problem change if the equation was $x^3 - 2x^2 - 3x + 2 = 0$?

Tip:

Always start by factoring out the greatest common factor (GCF) when simplifying polynomial equations!