1. **State the problem:** Simplify the expression $$3x^2y(x^3y - y^2x^2) - 5x^3y^2x^2 - 2xyx^4y + 5xy^2x^4y^2 - 4x^3uyy^2.$$ 2. **Rewrite and clarify terms:** Note that $x^3y$ means $x^3 \cdot y$, and similarly for other terms. Also, $уху^2$ likely means $u \cdot x \cdot y \cdot y^2 = uxy^3$. 3. **Distribute inside the parentheses:** $$3x^2y(x^3y) - 3x^2y(y^2x^2) - 5x^3y^2x^2 - 2xyx^4y + 5xy^2x^4y^2 - 4x^3uyy^2$$ 4. **Simplify each term:** - $3x^2y \cdot x^3y = 3x^{2+3}y^{1+1} = 3x^5y^2$ - $3x^2y \cdot y^2x^2 = 3x^{2+2}y^{1+2} = 3x^4y^3$ - $-5x^3y^2x^2 = -5x^{3+2}y^2 = -5x^5y^2$ - $-2xyx^4y = -2x^{1+4}y^{1+1} = -2x^5y^2$ - $5xy^2x^4y^2 = 5x^{1+4}y^{2+2} = 5x^5y^4$ - $-4x^3uyy^2 = -4x^3u y^{1+2} = -4x^3uy^3$ 5. **Rewrite the expression:** $$3x^5y^2 - 3x^4y^3 - 5x^5y^2 - 2x^5y^2 + 5x^5y^4 - 4x^3uy^3$$ 6. **Combine like terms:** - Combine $3x^5y^2 - 5x^5y^2 - 2x^5y^2 = (3 - 5 - 2)x^5y^2 = -4x^5y^2$ - Other terms remain as is. 7. **Final simplified expression:** $$-4x^5y^2 - 3x^4y^3 + 5x^5y^4 - 4x^3uy^3$$