Simplify the fraction (x^2-y^2+2x+1)/[(x+y)^2+2(x+y)+1].

We'll group the terms x^2, 2x, 1, from numerator, and
we'll create a perfect square:

x^2+2x+1 =
(x+1)^2

We'll re-write the
numerator:

(x+1)^2 - y^2

We've
get a difference of squares:

(x+1)^2 - y^2 =
(x+1-y)(x+1+y)

We'll analyze the denominator and we'll
notice that it is a perfect square, too.

[(x+y)^2+2(x+y)+1]
= (x + y + 1)^2

We'll re-write the
fraction:

(x + 1 - y)(x + 1 + y)/(x + y +
1)^2

We'll simplify by x + 1 + y and we'll
get:

(x^2-y^2+2x+1)/[(x+y)^2+2(x+y)+1]=(x+1-y)/(x+y+1)