Answer
$ \frac{2d}{a^{2}+ab+b^{2}}$
Work Step by Step
$(\frac{1}{a^{3}-b^{3}} . \frac{ac+ad-bc-bd}{1}) - \frac{c-d}{a^{2}+ab+b^{2}}$
$= (\frac{ac+ad-bc-bd}{a^{3}-b^{3}}) - \frac{c-d}{a^{2}+ab+b^{2}}$
$= (\frac{a(c+d)-b(c+d)}{a^{3}-b^{3}}) - \frac{c-d}{a^{2}+ab+b^{2}}$
$= (\frac{(a-b)(c+d)}{a^{3}-b^{3}}) - \frac{c-d}{a^{2}+ab+b^{2}}$
Factors of $a^{3}-b^{3}$ are $(a-b)(a^{2}+ab+b^{2})$
$= (\frac{(a-b)(c+d)}{(a-b)(a^{2}+ab+b^{2})}) - \frac{c-d}{a^{2}+ab+b^{2}}$
$= \frac{c+d}{a^{2}+ab+b^{2}} - \frac{c-d}{a^{2}+ab+b^{2}}$
$= \frac{c+d-c+d}{a^{2}+ab+b^{2}} $
$= \frac{2d}{a^{2}+ab+b^{2}} $
