1. **State the problem:** Simplify the expression $$\frac{x^2 - xy + y^2}{x - y} - \frac{x^2 + xy + y^2}{x + y}$$. 2. **Recall the formula and rules:** We want to simplify each rational expression and then subtract. Notice the numerators are quadratic expressions and denominators are linear binomials. 3. **Factor or rewrite the numerators if possible:** - The numerator $x^2 - xy + y^2$ cannot be factored easily over real numbers but can be related to $(x - y)^2$: $$x^2 - xy + y^2 = (x - y)^2 + xy$$ - The numerator $x^2 + xy + y^2$ similarly relates to $(x + y)^2$: $$x^2 + xy + y^2 = (x + y)^2 - xy$$ 4. **Rewrite the expression using these identities:** $$\frac{(x - y)^2 + xy}{x - y} - \frac{(x + y)^2 - xy}{x + y}$$ 5. **Split the fractions:** $$\frac{(x - y)^2}{x - y} + \frac{xy}{x - y} - \left(\frac{(x + y)^2}{x + y} - \frac{xy}{x + y}\right)$$ 6. **Simplify terms:** - $$\frac{(x - y)^2}{x - y} = x - y$$ - $$\frac{(x + y)^2}{x + y} = x + y$$ So the expression becomes: $$ (x - y) + \frac{xy}{x - y} - (x + y) + \frac{xy}{x + y} $$ 7. **Group like terms:** $$ (x - y - x - y) + \frac{xy}{x - y} + \frac{xy}{x + y} = -2y + xy\left(\frac{1}{x - y} + \frac{1}{x + y}\right) $$ 8. **Combine the fractions inside the parentheses:** $$ \frac{1}{x - y} + \frac{1}{x + y} = \frac{(x + y) + (x - y)}{(x - y)(x + y)} = \frac{2x}{x^2 - y^2} $$ 9. **Substitute back:** $$ -2y + xy \cdot \frac{2x}{x^2 - y^2} = -2y + \frac{2x^2 y}{x^2 - y^2} $$ 10. **Write the final simplified expression:** $$ \boxed{-2y + \frac{2x^2 y}{x^2 - y^2}} $$ This is the simplified form of the original expression.