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It seems like you uploaded an image of a math problem asking to factor a multivariable expression. The expression given is:

\[
-42xv^3z^3 + 12x^4v^4z^7
\]

To factor this expression completely, we need to follow these steps:

### 1. Find the greatest common factor (GCF):
- **Coefficients:** The GCF of -42 and 12 is 6 (specifically, -6 to factor the negative sign out).
- **Variables:**
   - For \( x \), the GCF is \( x \), since the smallest exponent of \( x \) in the terms is 1.
   - For \( v \), the GCF is \( v^3 \), since the smallest exponent of \( v \) is 3.
   - For \( z \), the GCF is \( z^3 \), since the smallest exponent of \( z \) is 3.

Thus, the GCF of the entire expression is \( -6xv^3z^3 \).

### 2. Factor out the GCF:
Now we factor out \( -6xv^3z^3 \) from each term:

\[
-6xv^3z^3(-7 + 2x^3v^1z^4)
\]

### Final Factored Expression:
\[
-6xv^3z^3(-7 + 2x^3vz^4)
\]

This is the fully factored form of the expression.

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Would you like more details or have any questions on this?

Here are some related questions for practice:
1. How do you find the greatest common factor for terms with variables?
2. How do you factor expressions involving multiple variables?
3. What is the importance of factoring negative signs in an expression?
4. How does factoring help simplify algebraic expressions?
5. Can the factored form of an expression be verified by expanding it back?

**Tip:** When factoring, always check each variable and constant separately to find the greatest common factor, and keep track of the signs carefully!

Factor completely: -42xv^3z^3 + 12x^4v^4z^7

Learn how to completely factor the multivariable expression -42xv^3z^3 + 12x^4v^4z^7 by finding the greatest common factor (GCF) and factoring it out. This step-by-step solution is suitable for high school students, covering core algebraic concepts and factoring methods.

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