Take the non-complex trinomial a²±2ab+b² and a²-b². These can be written as the perfect square (a±b)² and (a-b)(a+b).
So the square root of the perfect square trinomial is a±b, therefore a+b or a-b is always a factor of the difference of the same two squares a and b.
For a complex trinomial a²±2aib-b² and a²-b², the complex trinomial can be written as the perfect square (a±ib)². This can also be written a²-b²±2aib, so it’s clear that the difference of the two squares a²-b² is the real part of the trinomial.