let the two consecutive integers be:

x and x+1, where x is integer

the product of two consecutive integers is 5 more than their sum

x(x + 1) = 5 + x + (x + 1)

by solving we find

x1 = 3

x2 = -2

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3 + 1 = 4

indeed: 3*4 = 12 = 5 + 3 + 4

-2 + 1 = -1

indeed: (-2)*(-1) = 2 = 5 + (-2) + (-1)

there are two pairs of such integers: (3,4) and ((-2),(-1)).