1. **State the problem:** Simplify the expression $$\frac{y+1}{y^{2}-4} - \frac{5}{y+2}$$. 2. **Recall the formula and rules:** - Factor the denominator if possible. - Find a common denominator to combine the fractions. - Simplify the resulting expression. 3. **Factor the denominator:** $$y^{2}-4 = (y-2)(y+2)$$ 4. **Rewrite the expression with factored denominator:** $$\frac{y+1}{(y-2)(y+2)} - \frac{5}{y+2}$$ 5. **Find the common denominator:** The common denominator is $(y-2)(y+2)$. 6. **Rewrite the second fraction with the common denominator:** $$\frac{5}{y+2} = \frac{5(y-2)}{(y+2)(y-2)}$$ 7. **Combine the fractions:** $$\frac{y+1}{(y-2)(y+2)} - \frac{5(y-2)}{(y-2)(y+2)} = \frac{y+1 - 5(y-2)}{(y-2)(y+2)}$$ 8. **Simplify the numerator:** $$y+1 - 5(y-2) = y + 1 - 5y + 10 = -4y + 11$$ 9. **Final simplified expression:** $$\frac{-4y + 11}{(y-2)(y+2)}$$ This is the simplified form of the given expression.