The roots are a and b:
a + b = p and ab = 12
(a + b)^2 = p^2
(a - b)^2 = (a + b)^2 - 4ab
=> (a - b)^2 = p^2 - 12 * 4 = p^2 - 48
If |a - b| ≥ 12 { Difference between the roots is at least 12}
then, (a - b)^2 ≥ 144
p^2 - 48 ≥ 144
p^2 ≥ 192
P ≥ 8√3 or P ≤ -8√3

In keynote 4, example 2, total non-negative unordered integral solutions is just 1 (4,0) and total non-negative ordered integral solutions are 2 (4,0 and 0,4). Please Correct me if I'm wrong.

@vikas_saini can u plz explain 2nd and 3rd+4th step again. 2nd step:what is -1-what are we reducing in it
3rd step: 3rd+4th step doesnt giv value in 2nd step. what are we missing in it. 3rd+4th= 2T-2