Given the expression $x^3 - 3x^2 - 4x + 12$, we can factor it by grouping. Here’s the step-by-step process:

Step 1: Group the terms

Group the terms in pairs to factor by grouping:

$(x^3 - 3x^2) - (4x - 12)$

Step 2: Factor out the common factors in each group

• From the first group $(x^3 - 3x^2)$, factor out $x^2$: $x^2(x - 3)$
• From the second group $(-4x + 12)$, factor out $-4$: $-4(x - 3)$

Now, the expression becomes:

$x^2(x - 3) - 4(x - 3)$

Step 3: Factor out the common binomial factor

Both terms now have a common factor of $(x - 3)$, so we factor that out:

$(x - 3)(x^2 - 4)$

Step 4: Factor the difference of squares

Notice that $x^2 - 4$ is a difference of squares, which can be factored further:

$x^2 - 4 = (x - 2)(x + 2)$

Final Factored Form:

The fully factored form of the expression is:

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

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Would you like more details or have any questions? Here are some related questions:

1. How do you factor a cubic polynomial in general?
2. What is the difference of squares, and why is it useful?
3. How do you factor polynomials with higher degrees?
4. Can you apply synthetic division to factor cubic polynomials?
5. What are the possible real roots of a cubic equation?

Tip: When factoring higher-degree polynomials, always look for common factors or patterns like grouping or differences of squares. These techniques simplify the expression significantly.