1. **State the problem:** We need to find which expressions are equivalent to $$-6z^3 (2z + 10z^2)$$. 2. **Use the distributive property:** Multiply $$-6z^3$$ by each term inside the parentheses: $$-6z^3 \times 2z = -12z^{3+1} = -12z^4$$ $$-6z^3 \times 10z^2 = -60z^{3+2} = -60z^5$$ So, the expression simplifies to: $$-12z^4 - 60z^5$$ 3. **Check each option:** - Option 1: $$3z^3 (-4z + 20z^2)$$ Multiply: $$3z^3 \times -4z = -12z^{4}$$ $$3z^3 \times 20z^2 = 60z^{5}$$ So, this equals $$-12z^4 + 60z^5$$, which is **not** the same as the original. - Option 2: $$-12z^4 - 60z^5$$ This matches exactly the simplified original expression. - Option 3: $$12z^3 (-z - 5z^2)$$ Multiply: $$12z^3 \times -z = -12z^4$$ $$12z^3 \times -5z^2 = -60z^5$$ This equals $$-12z^4 - 60z^5$$, which matches the original. - Option 4: $$(-6z - 30z^2) 2z^3$$ Multiply: $$-6z \times 2z^3 = -12z^{4}$$ $$-30z^2 \times 2z^3 = -60z^{5}$$ This equals $$-12z^4 - 60z^5$$, which matches the original. 4. **Conclusion:** The equivalent expressions are: - $$-12z^4 - 60z^5$$ - $$12z^3 (-z - 5z^2)$$ - $$(-6z - 30z^2) 2z^3$$ **Final answer:** The expressions equivalent to $$-6z^3 (2z + 10z^2)$$ are options 2, 3, and 4.