1. **State the problem:** Simplify the expression $a^2b(a^3b - b^2a^2) + 4a^3b^2a^2 - 2aba^4b + 7ab^0a^4b^2 - 3a^3bab^2$. 2. **Recall exponent rules:** - $a^m a^n = a^{m+n}$ - $b^m b^n = b^{m+n}$ - $a^0 = 1$ 3. **Distribute and simplify each term:** - $a^2b(a^3b) = a^{2+3}b^{1+1} = a^5b^2$ - $a^2b(-b^2a^2) = -a^{2+2}b^{1+2} = -a^4b^3$ - $4a^3b^2a^2 = 4a^{3+2}b^2 = 4a^5b^2$ - $-2aba^4b = -2a^{1+4}b^{1+1} = -2a^5b^2$ - $7ab^0a^4b^2 = 7a^{1+4}b^{0+2} = 7a^5b^2$ - $-3a^3bab^2 = -3a^{3+1}b^{1+2} = -3a^4b^3$ 4. **Rewrite the expression with simplified terms:** $$a^5b^2 - a^4b^3 + 4a^5b^2 - 2a^5b^2 + 7a^5b^2 - 3a^4b^3$$ 5. **Group like terms:** - Terms with $a^5b^2$: $a^5b^2 + 4a^5b^2 - 2a^5b^2 + 7a^5b^2 = (1 + 4 - 2 + 7)a^5b^2 = 10a^5b^2$ - Terms with $a^4b^3$: $-a^4b^3 - 3a^4b^3 = (-1 - 3)a^4b^3 = -4a^4b^3$ 6. **Final simplified expression:** $$10a^5b^2 - 4a^4b^3$$