1. **State the problem:** Simplify the expression $$2ab^2 - [(a+b)(a-b) - ab(a - 2ab)]a^2b$$. 2. **Recall formulas and rules:** - Difference of squares: $$(a+b)(a-b) = a^2 - b^2$$ - Distributive property: multiply terms inside brackets. 3. **Simplify inside the brackets:** $$(a+b)(a-b) = a^2 - b^2$$ 4. **Simplify the term $ab(a - 2ab)$:** $$ab(a - 2ab) = ab \cdot a - ab \cdot 2ab = a^2b - 2a b^2 a b = a^2 b - 2 a^2 b^2$$ 5. **Substitute back into the bracket:** $$[(a+b)(a-b) - ab(a - 2ab)] = (a^2 - b^2) - (a^2 b - 2 a^2 b^2) = a^2 - b^2 - a^2 b + 2 a^2 b^2$$ 6. **Multiply the bracket by $a^2 b$:** $$[a^2 - b^2 - a^2 b + 2 a^2 b^2] a^2 b = a^2 b \cdot a^2 - a^2 b \cdot b^2 - a^2 b \cdot a^2 b + a^2 b \cdot 2 a^2 b^2$$ $$= a^4 b - a^2 b^3 - a^4 b^2 + 2 a^4 b^3$$ 7. **Rewrite the original expression:** $$2 a b^2 - (a^4 b - a^2 b^3 - a^4 b^2 + 2 a^4 b^3)$$ 8. **Distribute the minus sign:** $$2 a b^2 - a^4 b + a^2 b^3 + a^4 b^2 - 2 a^4 b^3$$ 9. **Final simplified expression:** $$\boxed{2 a b^2 - a^4 b + a^2 b^3 + a^4 b^2 - 2 a^4 b^3}$$