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$O(n^2 \cdot 2^s) = O(2^{\log_2(n^2)} \cdot 2^s) = O(2^{s+2log_2(n)})$, so your provided runtime is still in $EXPTIME$ as long as $n \in 2^{O(\text{poly}(s))}$. For your second question, I'm not ...