1. **State the problem:** Show that the expressions $$\frac{x^4 - y^4}{(x + y)^2} \times (x^2 - y^2)^2 \div \frac{x^2 + y^2}{x^2 + 2xy + y^2}$$ and $$(x + y)(3x - y) - 3(x - y)^2 - 4y(2x - y) + 1$$ simplify to the same expression but are not equivalent. 2. **Recall important formulas and identities:** - Difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ - Factorization of difference of fourth powers: $$x^4 - y^4 = (x^2 - y^2)(x^2 + y^2)$$ - Square of binomial: $$(a \pm b)^2 = a^2 \pm 2ab + b^2$$ 3. **Simplify the first expression:** Start with numerator of first fraction: $$x^4 - y^4 = (x^2 - y^2)(x^2 + y^2)$$ So, $$\frac{x^4 - y^4}{(x + y)^2} = \frac{(x^2 - y^2)(x^2 + y^2)}{(x + y)^2}$$ Multiply by $$(x^2 - y^2)^2$$: $$\frac{(x^2 - y^2)(x^2 + y^2)}{(x + y)^2} \times (x^2 - y^2)^2 = \frac{(x^2 - y^2)^3 (x^2 + y^2)}{(x + y)^2}$$ Divide by $$\frac{x^2 + y^2}{x^2 + 2xy + y^2}$$ which is equivalent to multiplying by its reciprocal: $$\times \frac{x^2 + 2xy + y^2}{x^2 + y^2}$$ So the entire expression is: $$\frac{(x^2 - y^2)^3 (x^2 + y^2)}{(x + y)^2} \times \frac{x^2 + 2xy + y^2}{x^2 + y^2} = \frac{(x^2 - y^2)^3 (x^2 + 2xy + y^2)}{(x + y)^2}$$ Since $$x^2 + 2xy + y^2 = (x + y)^2$$, substitute: $$= \frac{(x^2 - y^2)^3 (x + y)^2}{(x + y)^2}$$ Cancel $$(x + y)^2$$: $$= (x^2 - y^2)^3$$ Recall $$x^2 - y^2 = (x - y)(x + y)$$, so: $$= ((x - y)(x + y))^3 = (x - y)^3 (x + y)^3$$ 4. **Simplify the second expression:** Expand each term: - $$(x + y)(3x - y) = 3x^2 - xy + 3xy - y^2 = 3x^2 + 2xy - y^2$$ - $$-3(x - y)^2 = -3(x^2 - 2xy + y^2) = -3x^2 + 6xy - 3y^2$$ - $$-4y(2x - y) = -8xy + 4y^2$$ Sum all terms plus 1: $$3x^2 + 2xy - y^2 - 3x^2 + 6xy - 3y^2 - 8xy + 4y^2 + 1$$ Combine like terms: - $$3x^2 - 3x^2 = 0$$ - $$2xy + 6xy - 8xy = 0$$ - $$-y^2 - 3y^2 + 4y^2 = 0$$ So expression reduces to: $$1$$ 5. **Conclusion:** The first expression simplifies to: $$(x - y)^3 (x + y)^3$$ The second expression simplifies to: $$1$$ They simplify to different expressions, so they are not equivalent. However, the problem states they simplify to the same but are not equivalent, so let's verify the problem statement carefully. **Re-examining the problem:** The problem says "Show that the following expressions simplify to the same but are not equivalent." This likely means they simplify to the same expression after simplification steps but are not algebraically equivalent expressions (e.g., domain restrictions or intermediate forms differ). Since the second expression simplifies to 1, and the first to $$(x - y)^3 (x + y)^3$$, they are not the same. **Therefore, the first expression simplifies to $$(x - y)^3 (x + y)^3$$ and the second simplifies to 1, so they are not equivalent and do not simplify to the same expression.** **If the problem intended to show they simplify to the same, please verify the problem statement or expressions given.**