1. **State the problem:** Factor completely the expression $$-3x^6yz^2 + 2x^2y$$. 2. **Identify common factors:** Look for the greatest common factor (GCF) in each term. - Both terms have $x^2$ and $y$. 3. **Extract the GCF:** $$\text{GCF} = x^2y$$ 4. **Factor out the GCF:** $$-3x^6yz^2 + 2x^2y = x^2y\left(\frac{-3x^6yz^2}{x^2y} + \frac{2x^2y}{x^2y}\right)$$ 5. **Simplify inside the parentheses:** $$= x^2y\left(-3x^{6-2}z^2 + 2\right) = x^2y\left(-3x^4z^2 + 2\right)$$ 6. **Final factored form:** $$\boxed{x^2y(-3x^4z^2 + 2)}$$ This is the complete factorization since the second factor cannot be factored further over the integers.