Rút gọn

To simplify the given expression, let's break it down step by step:

1. Simplify the numerator:
(x^2 + xy + y^2)

2. Simplify the denominator:
(x^3 + 3xy * (x + y)^2 + y^3)

3. Simplify the expression inside the first set of parentheses:
(x - y)

4. Simplify the expression inside the second set of parentheses:
(x^2 + xy + y^2)

5. Simplify the expression inside the third set of parentheses:
(x - y)

Now, let's simplify the entire expression:

(x^2 + xy + y^2) / (x^3 + 3xy * (x + y)^2 + y^3) / ((x - y) / (x^2 + xy + y^2) - (x + y) / (x - y))

= (x^2 + xy + y^2) / (x^3 + 3xy * (x + y)^2 + y^3) * ((x - y) / (x^2 + xy + y^2) - (x + y) / (x - y))

= (x^2 + xy + y^2) * (x - y) / (x^3 + 3xy * (x + y)^2 + y^3) * (x - y) / (x^2 + xy + y^2) - (x + y) / (x - y)

= (x^2 + xy + y^2) * (x - y)^2 / (x^3 + 3xy * (x + y)^2 + y^3) * (x^2 + xy + y^2) - (x + y) / (x - y)

= (x^2 + xy + y^2) * (x - y)^2 / (x^3 + 3xy * (x + y)^2 + y^3) * (x^2 + xy + y^2) - (x + y) * (x - y) / (x - y)

= (x^2 + xy + y^2) * (x - y)^2 / (x^3 + 3xy * (x + y)^2 + y^3) * (x^2 + xy + y^2) - (x^2 - y^2) / (x - y)

= (x^2 + xy + y^2) * (x - y)^2 / (x^3 + 3xy * (x + y)^2 + y^3) * (x^2 + xy + y^2) - (x^2 - y^2) * (x - y) / (x - y)

= (x^2 + xy + y^2) * (x - y)^2 / (x^3 + 3xy * (x + y)^2 + y^3) * (x^2 + xy + y^2) - (x^3 - xy^2 - x^2y + y^3) / (x - y)

= (x^2 + xy + y^2) * (x - y)^2 / (x^3 + 3xy * (x + y)^2 + y^3) * (x^2 + xy + y^2) - (x^3 - xy^2 - x^2y + y^3) / (x - y)

= (x^2 + xy + y^2)^2 * (x - y)^2 - (x^3 - xy^2 - x^2y + y^3) / (x^3 + 3xy * (x + y)^2 + y^3)

= (x^4 + 2x^3y + x^2y^2 + 2x^3y - 2xy^3 + y^4) - (x^3 - xy^2 - x^2y + y^3) / (x^3 + 3xy * (x + y)^2 + y^3)

= x^4 + 4x^3y + x^2y^2 - 2xy^3 + y^4 - x^3 + xy^2 + x^2y - y^3 / (x^3 + 3xy * (x + y)^2 + y^3)

= x^4 + 4x^3y + x^2y^2 - 2xy^3 + y^4 - x^3 + xy^2 + x^2y - y^3 / x^3 + 3xy * (x + y)^2 + y^3

Therefore, the simplified expression is:

(x^4 + 4x^3y + x^2y^2 - 2xy^3 + y^4 - x^3 + xy^2 + x^2y - y^3) / (x^3 + 3xy * (x + y)^2 + y^3)