Could anyone please tell me if my steps and solution is correct or should I change something?
Compute the terms $c_2,c_3,c_4$ of a recursively defined relation
\begin{align}
c_k &=c_{k-1}+kc_{k-2}+1 & \text{ for all integers $k \ge 2$}
\end{align}
where initial conditions are $c_0=1$ and $c_1=2$.
Working: \begin{align} k = 2,\\ c_2 &= c_1 + 2c_0 + 1\\ &= 2 + 2(1) + 1\\ &= 5\\ \\ k = 3,\\ c_3 &= c_2 + 3c_1 +1\\ &= 5 + 3(2) +1\\ &= 12\\ \\ k = 4,\\ c_4 &= c_3 + 4c_2+1\\ &= 12 + 4(5) + 1\\ &= 33 \end{align}
Perfect.
You did exactly what you were asked to do with sufficient details.
If you want to spend more time on the problem, you may try to find a formula for $C_n$ in terms of $n$