1. Problem: Simplify the expression $\frac{1}{y^2+3y+2} \div \frac{2}{y^2-4}$. 2. Formula and rule: To divide fractions use the rule $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$. Important rules: Factor polynomials, cancel common nonzero factors, and note excluded values where denominators are zero. 3. Factor the quadratics. $$y^2+3y+2=(y+1)(y+2)$$ $$y^2-4=(y-2)(y+2)$$ 4. Convert the division to multiplication and substitute the factored forms. $$\frac{1}{(y+1)(y+2)} \div \frac{2}{(y-2)(y+2)} = \frac{1}{(y+1)(y+2)} \cdot \frac{(y-2)(y+2)}{2}$$ 5. Cancel the common factor $(y+2)$ showing the canceled factors. $$= \frac{1}{(y+1)\cancel{(y+2)}} \cdot \frac{(y-2)\cancel{(y+2)}}{2}$$ 6. Multiply the remaining factors and simplify. $$= \frac{y-2}{2(y+1)}$$ 7. Final answer and domain. Final simplified form: $\frac{y-2}{2(y+1)}$. Domain restrictions: $y \neq -2, -1, 2$ because these values make a denominator zero.