Answer
$ \frac{2d}{a^2+ab+b^2}$.
Work Step by Step
The given expression is
$\Rightarrow \left ( \frac{1}{a^3-b^3}\cdot \frac{ac+ad-bc-bd}{1} \right )-\frac{c-d}{a^2+ab+b^2}$
Factor each term in the bracket as shown below.
$\Rightarrow a^3-b^3$
Use the special formula $A^3-B^3=(A-B)(A^2+AB+B^2)$.
$\Rightarrow (a-b)(a^2+ab+b^2)$
and $\Rightarrow ac+ad-bc-bd$
Group terms.
$\Rightarrow (ac+ad)+(-bc-bd)$
Factor each group.
$\Rightarrow a(c+d)-b(c+d)$
Factor out $(c+d)$.
$\Rightarrow (c+d)(a-b)$
Back substitute the factors into the given expression.
$\Rightarrow \left ( \frac{1}{(a-b)(a^2+ab+b^2)}\cdot \frac{(c+d)(a-b)}{1} \right )-\frac{c-d}{a^2+ab+b^2}$
Cancel common terms.
$\Rightarrow \frac{c+d}{a^2+ab+b^2}-\frac{c-d}{a^2+ab+b^2}$
$\Rightarrow \frac{c+d-(c-d)}{a^2+ab+b^2}$
Simplify.
$\Rightarrow \frac{c+d-c+d}{a^2+ab+b^2}$
$\Rightarrow \frac{2d}{a^2+ab+b^2}$.
