Factor the following:
\[ -8x^2y^4z^3 + 12x^4yz^3.\]

To factor the expression:

\[
-8x^2y^4z^3 + 12x^4yz^3,
\]

we will follow these steps:

### Step 1: Find the greatest common factor (GCF).
We need to find the GCF of the coefficients and variables.

- For the coefficients, the GCF of \(-8\) and \(12\) is \(4\).
- For the \(x\) terms, the GCF of \(x^2\) and \(x^4\) is \(x^2\).
- For the \(y\) terms, the GCF of \(y^4\) and \(y\) is \(y\).
- For the \(z\) terms, both terms have \(z^3\), so the GCF is \(z^3\).

Thus, the overall GCF is \(4x^2yz^3\).

### Step 2: Factor out the GCF.

Now, we factor out \(4x^2yz^3\) from the expression:

\[
-8x^2y^4z^3 + 12x^4yz^3 = 4x^2yz^3 \left( \frac{-8x^2y^4z^3}{4x^2yz^3} + \frac{12x^4yz^3}{4x^2yz^3} \right).
\]

Simplifying each term inside the parentheses:

- \(\frac{-8x^2y^4z^3}{4x^2yz^3} = -2y^3\),
- \(\frac{12x^4yz^3}{4x^2yz^3} = 3x^2\).

So the factored form of the expression is:

\[
4x^2yz^3(-2y^3 + 3x^2).
\]

### Final Answer:
\[
4x^2yz^3(-2y^3 + 3x^2).
\]

Would you like further details on this process or have any questions?

Here are 5 related questions for further practice:
1. Factor \( -6x^2y^2z + 9x^3yz^2 \).
2. Simplify \( 15x^5y^3z^2 - 20x^2y^2z \).
3. Factor \( -4x^3yz + 8x^2y^4z^2 \).
4. What is the GCF of \( 18x^6y^2 \) and \( 24x^3y^4 \)?
5. Factor \( -10x^2y^2 + 5x^3y \).

**Tip:** Always start by identifying the GCF when factoring an expression.

Learn how to factor the expression -8x^2y^4z^3 + 12x^4yz^3 by identifying the greatest common factor (GCF). This detailed solution covers key algebraic concepts such as factoring and GCF. Suitable for students in Grades 8-10.

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