1. **State the problem:** Simplify the expression $$(x-2y)(y-3x)+(x-y)(x-3y)-(y-3x)(4x-5y).$$ 2. **Use the distributive property (FOIL) to expand each product:** - Expand $(x-2y)(y-3x)$: $$xy - 3x^2 - 2y^2 + 6xy = -3x^2 + 7xy - 2y^2$$ - Expand $(x-y)(x-3y)$: $$x^2 - 3xy - xy + 3y^2 = x^2 - 4xy + 3y^2$$ - Expand $(y-3x)(4x-5y)$: $$4xy - 5y^2 - 12x^2 + 15xy = -12x^2 + 19xy - 5y^2$$ 3. **Substitute the expansions back into the expression:** $$(-3x^2 + 7xy - 2y^2) + (x^2 - 4xy + 3y^2) - (-12x^2 + 19xy - 5y^2)$$ 4. **Simplify by combining like terms and distributing the minus sign:** $$-3x^2 + 7xy - 2y^2 + x^2 - 4xy + 3y^2 + 12x^2 - 19xy + 5y^2$$ 5. **Combine like terms:** - For $x^2$: $-3x^2 + x^2 + 12x^2 = 10x^2$ - For $xy$: $7xy - 4xy - 19xy = -16xy$ - For $y^2$: $-2y^2 + 3y^2 + 5y^2 = 6y^2$ 6. **Final simplified expression:** $$10x^2 - 16xy + 6y^2$$