1. **State the problem:** Reduce the expression $$a^2b(a^3b - b^2a^2) + 4a^3b^2a^2 - 2aba^4b + 7ab^0a^4b^2 - 3a^3bab^2$$ to a standard polynomial. 2. **Distribute and simplify each term:** - First term: $$a^2b \times a^3b = a^{2+3}b^{1+1} = a^5b^2$$ - First term second part: $$a^2b \times (-b^2a^2) = -a^{2+2}b^{1+2} = -a^4b^3$$ - Second term: $$4a^3b^2a^2 = 4a^{3+2}b^2 = 4a^5b^2$$ - Third term: $$-2aba^4b = -2a^{1+1+4}b^{1+1} = -2a^6b^2$$ - Fourth term: $$7ab^0a^4b^2 = 7a^{1+4}b^{0+2} = 7a^5b^2$$ - Fifth term: $$-3a^3bab^2 = -3a^{3+1+1}b^{1+2} = -3a^5b^3$$ 3. **Rewrite the expression with simplified terms:** $$a^5b^2 - a^4b^3 + 4a^5b^2 - 2a^6b^2 + 7a^5b^2 - 3a^5b^3$$ 4. **Group like terms:** - Terms with $$a^5b^2$$: $$a^5b^2 + 4a^5b^2 + 7a^5b^2 = (1 + 4 + 7)a^5b^2 = 12a^5b^2$$ - Terms with $$a^4b^3$$: $$-a^4b^3$$ - Terms with $$a^6b^2$$: $$-2a^6b^2$$ - Terms with $$a^5b^3$$: $$-3a^5b^3$$ 5. **Final simplified polynomial:** $$12a^5b^2 - a^4b^3 - 2a^6b^2 - 3a^5b^3$$