1. **State the problem:** Simplify the expression $$\frac{1}{y^{2} + 3y + 2} \div \frac{2}{y^{2} - 4}$$. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{1}{y^{2} + 3y + 2} \div \frac{2}{y^{2} - 4} = \frac{1}{y^{2} + 3y + 2} \times \frac{y^{2} - 4}{2}$$. 3. **Factor the quadratic expressions:** - Factor $y^{2} + 3y + 2$ as $(y + 1)(y + 2)$. - Factor $y^{2} - 4$ as a difference of squares: $(y - 2)(y + 2)$. 4. **Rewrite the expression with factors:** $$\frac{1}{(y + 1)(y + 2)} \times \frac{(y - 2)(y + 2)}{2}$$. 5. **Multiply the numerators and denominators:** $$\frac{1 \times (y - 2)(y + 2)}{(y + 1)(y + 2) \times 2} = \frac{(y - 2)(y + 2)}{2(y + 1)(y + 2)}$$. 6. **Cancel common factors:** The factor $(y + 2)$ appears in numerator and denominator, so cancel it: $$\frac{(y - 2)\cancel{(y + 2)}}{2(y + 1)\cancel{(y + 2)}}$$ 7. **Final simplified expression:** $$\frac{y - 2}{2(y + 1)}$$. **Answer:** $$\frac{y - 2}{2(y + 1)}$$