We have to simplify (x^2-y^2+2x+1)/[(x+y)^2+2(x+y)+1]

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

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

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

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

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

=> [(x + 1 - y)/(x + y + 1)

**The required result is (x - y + 1)/ (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)**