1. **State the problem:** We are given a recurrence relation $$q_n = n q_{n-1} + n^2 q_{n-2}$$ with initial conditions $$q_0 = 1, q_1 = 1.$$ We want to find values of $q_n$ for $n = 2, 3, 4, 5$. 2. **Calculate $q_2$:** Using the recurrence for $n=2$, $$q_2 = 2 imes q_1 + 2^2 imes q_0 = 2 imes 1 + 4 imes 1 = 2 + 4 = 6.$$ 3. **Calculate $q_3$:** For $n=3$, $$q_3 = 3 imes q_2 + 3^2 imes q_1 = 3 imes 6 + 9 imes 1 = 18 + 9 = 27.$$ 4. **Calculate $q_4$:** For $n=4$, $$q_4 = 4 imes q_3 + 4^2 imes q_2 = 4 imes 27 + 16 imes 6 = 108 + 96 = 204.$$ 5. **Calculate $q_5$:** For $n=5$, $$q_5 = 5 imes q_4 + 5^2 imes q_3 = 5 imes 204 + 25 imes 27 = 1020 + 675 = 1695.$$ **Final answers:** $$q_2 = 6, \quad q_3 = 27, \quad q_4 = 204, \quad q_5 = 1695.$$