To simplify the given expression, we start by analyzing each term individually and looking for common patterns or identities that we can utilize. The expression is:
[tex]\[
\frac{2 b}{(a+b)^2(a-b)} - \frac{1}{a^2 - b^2} + \frac{1}{a^2 + 2ab + b^2}
\][/tex]
First, let's rewrite each term by recognizing possible factorizations:
#### Term 1:
[tex]\[
\frac{2 b}{(a+b)^2(a-b)}
\][/tex]
#### Term 2:
[tex]\[
\frac{1}{a^2 - b^2}
\][/tex]
Notice that [tex]\(a^2 - b^2\)[/tex] can be factored using the difference of squares:
[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]
So the second term becomes:
[tex]\[
\frac{1}{a^2 - b^2} = \frac{1}{(a+b)(a-b)}
\][/tex]
#### Term 3:
[tex]\[
\frac{1}{a^2 + 2ab + b^2}
\][/tex]
Notice that [tex]\(a^2 + 2ab + b^2\)[/tex] is a perfect square:
[tex]\[
a^2 + 2ab + b^2 = (a + b)^2
\][/tex]
So the third term becomes:
[tex]\[
\frac{1}{a^2 + 2ab + b^2} = \frac{1}{(a+b)^2}
\][/tex]
Now we substitute the factored forms back into the original expression:
[tex]\[
\frac{2 b}{(a+b)^2(a-b)} - \frac{1}{(a+b)(a-b)} + \frac{1}{(a+b)^2}
\][/tex]
Next, let's find a common denominator for these fractions. The common denominator will be [tex]\((a+b)^2(a-b)\)[/tex].
Rewriting each fraction with the common denominator:
[tex]\[
\frac{2 b}{(a+b)^2(a-b)} - \frac{(a+b)}{(a+b)^2(a-b)} + \frac{(a-b)}{(a+b)^2(a-b)}
\][/tex]
Now the expression is:
[tex]\[
\frac{2 b - (a+b) + (a-b)}{(a+b)^2(a-b)}
\][/tex]
Simplify the numerator:
[tex]\[
2 b - (a + b) + (a - b) = 2b - a - b + a - b = 0
\][/tex]
So, the expression simplifies to:
[tex]\[
\frac{0}{(a+b)^2(a-b)} = 0
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{0}
\][/tex]
