Let's factorize the given expression:

$3x^3 + 6x^2 - 3x - 6$

Step 1: Group the terms

We can group the terms to make factoring easier:

$(3x^3 + 6x^2) - (3x + 6)$

Step 2: Factor out the greatest common factor (GCF) from each group

• From the first group $3x^3 + 6x^2$, the GCF is $3x^2$.
• From the second group $-3x - 6$, the GCF is $-3$.

Now, factor the GCF out from both groups:

$3x^2(x + 2) - 3(x + 2)$

Step 3: Factor out the common binomial factor

Now we can factor out the common factor $(x + 2)$:

$(3x^2 - 3)(x + 2)$

Step 4: Simplify further

Notice that $3x^2 - 3$ has a common factor of 3:

$3(x^2 - 1)(x + 2)$

Finally, factor $x^2 - 1$ as a difference of squares:

$3(x - 1)(x + 1)(x + 2)$

Final Answer:

$3(x - 1)(x + 1)(x + 2)$

Would you like further details or have any questions?

Here are five related questions you might find interesting:

1. How do you factor a cubic polynomial?
2. What is the difference of squares method in factoring?
3. Can all polynomials be factored by grouping?
4. What are other methods of factoring polynomials?
5. How does the greatest common factor help in simplifying expressions?

Tip: Always check for the greatest common factor first when factoring expressions—it simplifies the process!