The difference of two squares occurs when the middle term or "b" of your polynomial is absent.
Looking at the standard form of
ax2+bx+c = 0
The "b" term is absent when the sum of the cross multiplication is zero:
b = (a*c) + (a*-c) = 0
so the the "c" term would have to be the product of a single number, one positive and one negative.
In other words, and using a = 1 for simplicity
x2-c2 = 0
to show you the series using small integers:
if c =1 then x2-1 = 0 factors to (x+1)(x-1) = 0
if c =2 then x2-4 = 0 factors to (x+2)(x-2) = 0
if c = 3 then x2 - 9 = 0 factors to (x+3)(x-3) = 0 and so on.
Notice how the signs alternate so that you are squaring the number by its positive and negative value. That means that when you multiply it out the cross product sum or middle term "b" has equal and opposite values and will then drop out of the expression.
(x+c)(x-c) = x2 + cx - cx - c2 = x2 - c2
The sum of two squares involves complex conjugates using the imaginary number or operator "i"
"i" is used to describe a number that satisfies the equation x2 = -1 so that x = √-1 = i
Every complex number has a corresponding conjugate and exist in pairs, this is typically written as
(a ± bi) but to use the notation from the example above i'll stick with x and c
(x+ci)(x-ci) = 0
multiplying that out yields
x2 + cxi - cxi - c2i2 = 0
Now knowing that i2 = -1 you get the answer you are looking for
x2 + c2 = 0
So for a positive integer series of c
where c = 1 then x2 + 1 = 0 factors to (x+i)(x-i)
if c =2 then x2 + 4 = 0 factors out to (x+2i)(x-2i)
if c = 3 then x2 + 9 = 0 factors out to (x+3i)(x-3i)
Again notice how the signs alternate so that you have one positive value and one negative value.