How can I calculate the maximum sum/product of sequence? I am looking for an algorithm in $O(N^2)$ that finds the maximum value that be obtained from a sequence of real numbers greater than 0 (e.g. $\{ 1, 2, 3 , 4\}$) by inserting a plus ($+$) or multiplication ($\times$) between the elements.

Really not having much luck and wondering if anyone has any insight on this?

It must follow normal order of precedence.

For example with $\{1,2,3,4\}$, the maximum is $1+2\times3\times4 = 25$ 
not $1\times2\times3\times4 = 24$ and not $1\times2+3+4 = 9$.
Let's say all numbers are positive (i.e. $\geq0$). Before and after $0$s you should always use $+$. This could also solved in $O(n^2)$ by first finding all zeros and then find maximum value for each group separated by zeros. Inside each group, for positive numbers, you should always use $\times$ except in $1$ cases, if ones are in the middle of a group, still you should choose $\times$. But when ones are at the very beginning or at the very end of the group, you should choose $+$.