1. **State the problem:** Simplify the expression \[ \frac{y + 1}{y^2 - 4} - \frac{y + 2}{5} \]. 2. **Recall the formula and rules:** - To subtract fractions, find a common denominator. - Factor expressions where possible. 3. **Factor the denominator:** $$y^2 - 4 = (y - 2)(y + 2)$$ 4. **Rewrite the expression:** $$\frac{y + 1}{(y - 2)(y + 2)} - \frac{y + 2}{5}$$ 5. **Find the common denominator:** The denominators are $(y - 2)(y + 2)$ and $5$. The common denominator is $$5(y - 2)(y + 2)$$. 6. **Rewrite each fraction with the common denominator:** $$\frac{y + 1}{(y - 2)(y + 2)} = \frac{5(y + 1)}{5(y - 2)(y + 2)}$$ $$\frac{y + 2}{5} = \frac{(y + 2)(y - 2)(y + 2)}{5(y - 2)(y + 2)}$$ Note: The second numerator is $$(y + 2)(y - 2)(y + 2) = (y + 2)^2(y - 2)$$. 7. **Combine the fractions:** $$\frac{5(y + 1) - (y + 2)^2(y - 2)}{5(y - 2)(y + 2)}$$ 8. **Expand the numerator:** First, expand $$(y + 2)^2 = y^2 + 4y + 4$$. Then multiply by $$(y - 2)$$: $$ (y^2 + 4y + 4)(y - 2) = y^3 - 2y^2 + 4y^2 - 8y + 4y - 8 = y^3 + 2y^2 - 4y - 8 $$ 9. **Substitute back:** $$5(y + 1) - (y^3 + 2y^2 - 4y - 8) = 5y + 5 - y^3 - 2y^2 + 4y + 8$$ 10. **Simplify numerator:** $$-y^3 - 2y^2 + 9y + 13$$ 11. **Final simplified expression:** $$\frac{-y^3 - 2y^2 + 9y + 13}{5(y - 2)(y + 2)}$$ This is the simplified form of the original expression. **Answer:** $$\boxed{\frac{-y^3 - 2y^2 + 9y + 13}{5(y - 2)(y + 2)}}$$