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