You can do it without trial and error by setting the first integer to x. Since the other integers are consecutive, the next number would be x +1, next would be x + 2, and then finally x +3
Add them together and set them to equal to -26, then solve for x.
x + (x+1) + (x+2) + (x+3)= -26
4x + 6 = -26
4x = -32
x = -8
the first number is -8, so x+1 is -7, x+2 is -6 and x + 3 is -5.
So your consecutive integers are -8,-7,-6, and -5, which when added together gives you -26.