The two consecutive integers can be written as:

x and x+1, where x is integer

The product of two consecutive integers is 19 more than their sum:

x(x + 1) = 19 + x + (x + 1)

by solving we find:

x1 = 5

x2 = -4

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for x1 = 5

5 + 1 = 6

for x2 = -4

-4 + 1 = -3

there are two solutions:

first solution: the numbers are 5 and 6

second solution: the numbers are -4 and -3