http://prntscr.com/l8rzbo
\(x^2 + m x + n = 0,~(m,n)\in \mathbb{Z} \\ \text{the only possible value for x is }x = -3\\ x^2 + m x + n = (x+3)^2 = x^2 + 6x + 9\\ m = 6\)
Since x = -3 is the only value that makes the quadratic = 0.....then this has a "double" root
So...the sum of the roots = -3 + -3 = -6
And the sum of the roots in the form ax^2 + bx + c = -b/a
So...in our case, the sum of the roots are = - m / 1 = -m
So
-m = -6
m = 6