1. **State the problem:** Simplify the expression $$\frac{X^2 - Y^2}{X + Y} \times \frac{X + 4Y}{2X^2 - XY - Y^2}$$. 2. **Recall formulas and rules:** - Difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$. - Factor quadratic expressions by grouping or using the quadratic formula. 3. **Factor the numerator and denominator:** - Numerator of first fraction: $$X^2 - Y^2 = (X - Y)(X + Y)$$. - Denominator of second fraction: $$2X^2 - XY - Y^2$$. 4. **Factor the quadratic in the denominator of the second fraction:** Find two numbers that multiply to $$2 \times (-1) = -2$$ and add to $$-1$$ (coefficient of $$X Y$$). These numbers are $$-2$$ and $$1$$. Rewrite: $$2X^2 - XY - Y^2 = 2X^2 - 2XY + XY - Y^2$$ Group: $$= (2X^2 - 2XY) + (XY - Y^2)$$ Factor each group: $$= 2X(X - Y) + Y(X - Y)$$ Factor out $$(X - Y)$$: $$= (X - Y)(2X + Y)$$. 5. **Rewrite the original expression with factored forms:** $$\frac{(X - Y)(X + Y)}{X + Y} \times \frac{X + 4Y}{(X - Y)(2X + Y)}$$ 6. **Cancel common factors:** - Cancel $$(X + Y)$$ in numerator and denominator. - Cancel $$(X - Y)$$ in numerator and denominator. Intermediate step showing cancellation: $$\frac{\cancel{(X - Y)}\cancel{(X + Y)}}{\cancel{(X + Y)}} \times \frac{X + 4Y}{\cancel{(X - Y)}(2X + Y)} = \frac{1 \times (X + 4Y)}{1 \times (2X + Y)} = \frac{X + 4Y}{2X + Y}$$ 7. **Final simplified expression:** $$\boxed{\frac{X + 4Y}{2X + Y}}$$