1. **State the problem:** Simplify the expression $$2ab^2 \cdot (a^2 - 2ab - 3b^2) - 2ab \cdot (a^2b - 2ab^2 - 6b^3)$$. 2. **Distribute each term:** $$2ab^2 \cdot a^2 = 2a^3b^2$$ $$2ab^2 \cdot (-2ab) = -4a^2b^3$$ $$2ab^2 \cdot (-3b^2) = -6ab^4$$ $$-2ab \cdot a^2b = -2a^3b^2$$ $$-2ab \cdot (-2ab^2) = +4a^2b^3$$ $$-2ab \cdot (-6b^3) = +12ab^4$$ 3. **Rewrite the expression with distributed terms:** $$2a^3b^2 - 4a^2b^3 - 6ab^4 - 2a^3b^2 + 4a^2b^3 + 12ab^4$$ 4. **Combine like terms:** $$\left(2a^3b^2 - 2a^3b^2\right) + \left(-4a^2b^3 + 4a^2b^3\right) + \left(-6ab^4 + 12ab^4\right)$$ 5. **Simplify each group:** $$0 + 0 + 6ab^4 = 6ab^4$$ **Final answer:** $$6ab^4$$