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The given expression is:

\[
3v^4 - 3v^4w^4
\]

To factor this expression completely, we begin by factoring out the common factor.

### Step 1: Factor out the greatest common factor (GCF)

The greatest common factor (GCF) between the terms \(3v^4\) and \(3v^4w^4\) is \(3v^4\). So, we can factor that out:

\[
3v^4(1 - w^4)
\]

### Step 2: Factor the difference of squares

The term inside the parentheses is \(1 - w^4\), which is a difference of squares. We can factor it using the formula for the difference of squares:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, \(a = 1\) and \(b = w^2\), so:

\[
1 - w^4 = (1 - w^2)(1 + w^2)
\]

### Step 3: Write the complete factorization

Substituting this back into the expression, we get:

\[
3v^4(1 - w^2)(1 + w^2)
\]

Finally, we observe that \(1 - w^2\) is itself a difference of squares, so we can factor that further:

\[
1 - w^2 = (1 - w)(1 + w)
\]

Thus, the complete factorization of the expression is:

\[
3v^4(1 - w)(1 + w)(1 + w^2)
\]

This is the fully factored form.

Would you like more details on any step? 

Here are five related questions to consider:
1. How do you identify a greatest common factor in a polynomial?
2. What is the difference of squares formula, and how is it applied?
3. Can the sum of squares \(1 + w^2\) be factored further?
4. How can we apply the same factoring techniques to higher-degree polynomials?
5. How does factoring help in solving polynomial equations?

**Tip:** Always check for common factors before applying other factoring techniques, as it simplifies the process significantly.

Learn how to factor the expression 3v^4 - 3v^4w^4 completely. This detailed step-by-step solution covers factoring out the greatest common factor (GCF), applying the difference of squares formula, and understanding how to fully factor complex expressions. Suitable for students in Grades 10-12.

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