x^2+mx+24= (x+P)(x+q)= x^2+x(p+q)+pq
so we can write m= p+q and pq=24
(p-q)^2 = (p+q)^2-4pq
now according to question, "the minimum value for the difference between p and q" can be p= -24 and q=-1 [as pq=24= (-24)*(-1)]
So, (p-q)^2 = (p+q)^2-4pq = [-24-1]^2-96 = 625-96 = 529
So, p-q = +/-23, now if we plug p=-24 and q=-1 we cannot take the positive value 23, because p-q= -24-(-1)=-23
So, -23 is our answer.
As it is written that "the minimum value for the difference between p and q" that's why we can't take 24 = 12*2 or 8*3 or 6*4.

Traps everywhere!
Be very careful:
We are told to min (p-q). We are not told if p is greater than q or not. This opens the possibility for negative values.
Since negative values are allowed:
24=24*1
So we could let q=24, p=1. Which would yield a value of -23. Any other value would be higher - ex 24=12*2, p=2, q=12 -> -10

We can see that p*q = 24 by expanding.the values of X , 24 can be arrived at by either (-12, -2), (-8,-3 )or (-6,-4) and (-1,-24). The minimum difference pair in this is (p-q)=-24+1=-23