Two integers, a and b have product of -48 what is the greatest possible sum of a and b.

The factors of 48 are:
1 x 48
2 x 24
3 x 16
4 x 12
6 x 8
and visa versa:   48 x 1  etc.

To make the product negative, one of the factors has to be positive and one has to be negative.  To find the greatest sum, make the bigger number the positive factor and the smaller number the negative factor.  So, -1 x 48 = -48 and the sum of -1 + 48 = 47 which would be the greatest possible sum.