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 values of $K$ in the interval $0 < K \leq 5$. 2. **Polynomial and roots:** The polynomial is quartic (degree 4), and its roots depend on the parameter $K$. Finding roots analytically for quartics with parameters can be complex, so typically numerical methods or software are used. 3. **Approach:** For each $K$ in $0 < K \leq 5$, substitute $K$ into the polynomial and solve $$s^4 + 2s^3 + (3+K)s^2 + (1+K)s + (1+K) = 0$$ for $s$. 4. **Example for $K=1$:** $$s^4 + 2s^3 + 4s^2 + 2s + 2 = 0$$ Use numerical methods (e.g., Newton-Raphson, or software) to find roots. 5. **Summary:** Roots vary with $K$. For each $K$, solve the quartic equation numerically to find the roots. Since the problem asks for roots for all $K$ in $0 < K \leq 5$, this is a parametric root-finding problem best handled computationally. **Final note:** The polynomial is $$y = s^4 + 2s^3 + (3+K)s^2 + (1+K)s + (1+K)$$ with $K$ in $(0,5]$; roots depend on $K$ and must be found numerically for each $K$.