1. **State the problem:** We are given two functions $S(s) = s^2 + s + 1$ and $T(s) = s - 5$. We need to find the sum $S(s) + T(s)$. 2. **Formula used:** To add two functions, we add their corresponding expressions: $$ (S + T)(s) = S(s) + T(s) $$ 3. **Add the functions:** $$ S(s) + T(s) = (s^2 + s + 1) + (s - 5) $$ 4. **Combine like terms:** $$ s^2 + s + 1 + s - 5 = s^2 + (s + s) + (1 - 5) $$ $$ = s^2 + 2s - 4 $$ 5. **Final answer:** $$ S(s) + T(s) = s^2 + 2s - 4 $$ This is the sum of the two functions.