We learn to recognize perfect square
trinomials as the form:

A²±2AB+B² = (A±B)²

The middle term in a perfect square trinomial
is twice the product of the square roots of 
the first and third terms. 

We look at our expression to factor:

x²-y²+2y-1

We recognize that the last three terms, 
-y²+2y-1, would form a perfect square trinomial 
if all the signs were changed. 

So we factor -1 out the last three
terms, for that will cause the signs to change:

x²-1(y²-2y+1)

No need to write the 1:

x²-(y²-2y+1)

Factor the perfect square trinomial

x²-(y-1)²

That's the difference of two squares:

[x-(y-1)][x+(y-1)]

Remove the inner parentheses

[x-y+1][x+y-1]

Change brackets to parentheses:

(x-y+1)(x+y-1)

Edwin