1. **State the problem:** Simplify the expression $$-\sqrt[3]{2x^{2}y^{2}} \cdot 2\sqrt[3]{15x^{5}y}$$. 2. **Recall the property of cube roots:** The product of cube roots can be combined as $$\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}$$. 3. **Rewrite the expression:** $$-\sqrt[3]{2x^{2}y^{2}} \cdot 2\sqrt[3]{15x^{5}y} = -2 \cdot \sqrt[3]{2x^{2}y^{2}} \cdot \sqrt[3]{15x^{5}y}$$ 4. **Combine the cube roots:** $$-2 \cdot \sqrt[3]{(2x^{2}y^{2})(15x^{5}y)} = -2 \cdot \sqrt[3]{30x^{7}y^{3}}$$ 5. **Simplify inside the cube root:** $$30x^{7}y^{3} = 30 \cdot x^{6} \cdot x^{1} \cdot y^{3}$$ 6. **Use the cube root of powers:** $$\sqrt[3]{x^{6}} = x^{2}$$ and $$\sqrt[3]{y^{3}} = y$$ 7. **Extract these from the cube root:** $$-2 \cdot x^{2} y \cdot \sqrt[3]{30 x^{1}} = -2 x^{2} y \sqrt[3]{30x}$$ **Final answer:** $$-2 x^{2} y \sqrt[3]{30x}$$