1. **State the problem:** Simplify the expression $$-5x^2 - 9y^3 + 3xy = - 2x^2 + 11y^3 - fxy$$ and find the value of $f$ such that both sides are equal. 2. **Collect like terms:** Move all terms to one side: $$-5x^2 - 9y^3 + 3xy + 2x^2 - 11y^3 + fxy = 0$$ 3. **Combine like terms:** $$(-5x^2 + 2x^2) + (-9y^3 - 11y^3) + (3xy + fxy) = 0$$ $$-3x^2 - 20y^3 + (3 + f)xy = 0$$ 4. **Set coefficients to zero:** For the equality to hold for all $x,y$, Coefficient of $xy$: $3 + f = 0 \\ \Rightarrow f = -3$ --- 5. **Next problem:** Simplify: $$5a^2b - 14ab^2 - 3 + 2a^2b - 6ab^2 - 9$$ 6. **Combine like terms:** $$(5a^2b + 2a^2b) + (-14ab^2 - 6ab^2) + (-3 - 9)$$ $$7a^2b - 20ab^2 - 12$$ --- 7. **Next problem:** Simplify $$(2x + 2) + (-3x - 5) + (2x + 9)$$ 8. **Combine like terms:** $$(2x - 3x + 2x) + (2 - 5 + 9)$$ $$1x + 6 = x + 6$$ --- 9. **Next problem:** Simplify $$(-3a^2 - b) - (-4a^2 + 6b) - (2a^2 - 4b)$$ 10. **Distribute minus signs:** $$-3a^2 - b + 4a^2 - 6b - 2a^2 + 4b$$ 11. **Combine like terms:** $$(-3a^2 + 4a^2 - 2a^2) + (-b -6b + 4b)$$ $$-1a^2 - 3b = -a^2 - 3b$$ --- 12. **Next problem:** Simplify $$(3x^4y^3 + 2x^3y^4 - 2xyd^2) - (-5x^6y^4 + 9x^4y^3 + xy^5)$$ 13. **Distribute minus:** $$3x^4y^3 + 2x^3y^4 - 2xyd^2 + 5x^6y^4 - 9x^4y^3 - xy^5$$ 14. **Combine like terms:** $$(3x^4y^3 - 9x^4y^3) + 2x^3y^4 + 5x^6y^4 - 2xyd^2 - xy^5$$ $$-6x^4y^3 + 2x^3y^4 + 5x^6y^4 - 2xyd^2 - xy^5$$ --- 15. **Last problem:** Simplify $$(6x + 2y - 3) + (-7x + xy - 5) - (-2x - 4y - 2)$$ 16. **Distribute minus and combine:** $$6x + 2y - 3 - 7x + xy - 5 + 2x + 4y + 2$$ 17. **Group like terms:** $$(6x - 7x + 2x) + (2y + 4y) + xy + (-3 - 5 + 2)$$ $$1x + 6y + xy - 6 = x + 6y + xy - 6$$