1. **State the problem:** We need to find the roots of the polynomial $$s^4 + 2s^3 + (3+K)s^2 + (1+K)s + (1+K)$$ for real values of $K$ in the range $0 \leq K \leq 5$. 2. **Polynomial and roots:** The polynomial is quartic (degree 4) with coefficients depending on $K$. Roots are values of $s$ such that the polynomial equals zero. 3. **Approach:** For each fixed $K$, solve $$s^4 + 2s^3 + (3+K)s^2 + (1+K)s + (1+K) = 0.$$ Since $K$ varies continuously, roots will vary continuously. 4. **Example calculations:** - For $K=0$, polynomial is $$s^4 + 2s^3 + 3s^2 + s + 1 = 0.$$ - For $K=1$, polynomial is $$s^4 + 2s^3 + 4s^2 + 2s + 2 = 0.$$ 5. **Root calculation method:** Use numerical methods (e.g., Newton-Raphson, or software) to find roots for each $K$. 6. **Summary:** Roots depend on $K$ and must be computed numerically for each $K$ in $[0,5]$. Since explicit closed-form roots for quartics with parameters are complicated, numerical root-finding is recommended for each $K$ value.