Is NSPACE(2^O(n)) = NSPACE(n^2 * 2^(O(n))
Yes. Clearly NSPACE($2^{O(n)}$) $\subseteq $  NSPACE($n^2 \cdot 2^{O(n)}$).
To show that NSPACE($2^{O(n)}$) $\supseteq $  NSPACE($n^2 \cdot 2^{O(n)}$) it suffices to notice that $n^2  \cdot 2^{O(n)} = 2^{O(n) + 2\log n}$ and $O(n)+2\log n =O(n)$.