No, you can't do it using integers. But the question doesn't restrict you to integers anywhere. It simply asks what 'x' produces the same output from both machines.
The upper machine produces x^2+2.
The lower machine produces (x+2)^2.
If the outputs are the same, then
x^2+2 = (x+2)^2 .
Crunch THAT through your own solve-o-matic machine, and you find that it works when x= -1/2 .