The missing word here is "average." For a life of simplified computations, when you hear the word "sum" you want to think "average" (and vice versa).
We can use the cross-multiplied average formula:
sum of items = (average of items)(number of items).
Let's apply it to the fact that the sum of the first 4 integers is 656:
656 = (average of first 4 numbers)(4)
average of first 4 numbers = 656/4 = 164
If this is the average of consecutive odd integers, then the integers must be 161, 163, 165, and 167, which balance in pairs around 164.
Therefore, the last four numbers in the sequence must be 169, 171, 173, 175. We could add these up to find the answer... or, we could use the average formula to find their sum!
They are balanced around 172, so that is their average, and there are four of them, so sum of second 4 numbers = (172)(4) = 400 + 280 + 8 = 688.
The correct answer is A.