1. **State the problem:** Reduce the expression $$3x^2y(x^3y - y^2x^2) - 5x^3y^2x^2 - 2xyx^4y + 5xy^2x^4y^2 - 4x^3yxy^2$$ to a standard polynomial. 2. **Expand the terms:** $$3x^2y(x^3y) - 3x^2y(y^2x^2) - 5x^3y^2x^2 - 2xyx^4y + 5xy^2x^4y^2 - 4x^3yxy^2$$ 3. **Simplify each term:** - $$3x^2y \cdot x^3y = 3x^{2+3}y^{1+1} = 3x^5y^2$$ - $$3x^2y \cdot y^2x^2 = 3x^{2+2}y^{1+2} = 3x^4y^3$$ - $$-5x^3y^2x^2 = -5x^{3+2}y^2 = -5x^5y^2$$ - $$-2xyx^4y = -2x^{1+4}y^{1+1} = -2x^5y^2$$ - $$5xy^2x^4y^2 = 5x^{1+4}y^{2+2} = 5x^5y^4$$ - $$-4x^3yxy^2 = -4x^{3+1}y^{1+2} = -4x^4y^3$$ 4. **Rewrite the expression with simplified terms:** $$3x^5y^2 - 3x^4y^3 - 5x^5y^2 - 2x^5y^2 + 5x^5y^4 - 4x^4y^3$$ 5. **Group like terms:** - Terms with $$x^5y^2$$: $$3x^5y^2 - 5x^5y^2 - 2x^5y^2$$ - Terms with $$x^4y^3$$: $$-3x^4y^3 - 4x^4y^3$$ - Term with $$x^5y^4$$: $$5x^5y^4$$ 6. **Combine like terms:** - $$3 - 5 - 2 = \cancel{3} - \cancel{5} - 2 = -4$$ so $$-4x^5y^2$$ - $$-3 - 4 = -7$$ so $$-7x^4y^3$$ - $$5x^5y^4$$ remains as is. 7. **Final simplified polynomial:** $$-4x^5y^2 - 7x^4y^3 + 5x^5y^4$$