1. **State the problem:** Simplify the expression $$(x-2y)(y-3x)+(x-y)(x-3y)-(y-3x)(4x-5y).$$ 2. **Recall the distributive property:** To simplify, we expand each product using $$(a+b)(c+d) = ac + ad + bc + bd$$ and then combine like terms. 3. **Expand each product:** - $$(x-2y)(y-3x) = x\cdot y + x\cdot(-3x) + (-2y)\cdot y + (-2y)\cdot(-3x) = xy - 3x^2 - 2y^2 + 6xy = -3x^2 + 7xy - 2y^2$$ - $$(x-y)(x-3y) = x\cdot x + x\cdot(-3y) + (-y)\cdot x + (-y)\cdot(-3y) = x^2 - 3xy - xy + 3y^2 = x^2 - 4xy + 3y^2$$ - $$(y-3x)(4x-5y) = y\cdot 4x + y\cdot(-5y) + (-3x)\cdot 4x + (-3x)\cdot(-5y) = 4xy - 5y^2 - 12x^2 + 15xy = -12x^2 + 19xy - 5y^2$$ 4. **Substitute expansions back into the expression:** $$(-3x^2 + 7xy - 2y^2) + (x^2 - 4xy + 3y^2) - (-12x^2 + 19xy - 5y^2)$$ 5. **Simplify by combining like terms:** $$-3x^2 + 7xy - 2y^2 + x^2 - 4xy + 3y^2 + 12x^2 - 19xy + 5y^2$$ Group 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:** $$10x^2 - 16xy + 6y^2$$ This is the simplified form of the original expression.