$$\frac{ { a }^{ -3 } + { b }^{ -3 } }{ { a }^{ 2 } -ab+ { b }^{ 2 } } \times { a }^{ 3 } { b }^{ 3 } - \frac{ { a }^{ 2 } - { b }^{ 2 } }{ a-b }$$
$0$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}}{a^{2}-ab+b^{2}}b^{3}-\frac{a^{2}-b^{2}}{a-b}$$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}b^{3}}{a^{2}-ab+b^{2}}-\frac{a^{2}-b^{2}}{a-b}$$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}b^{3}}{a^{2}-ab+b^{2}}-\frac{\left(a+b\right)\left(a-b\right)}{a-b}$$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}b^{3}}{a^{2}-ab+b^{2}}-\left(a+b\right)$$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}b^{3}}{a^{2}-ab+b^{2}}-a-b$$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}b^{3}}{a^{2}-ab+b^{2}}+\frac{\left(-a-b\right)\left(a^{2}-ab+b^{2}\right)}{a^{2}-ab+b^{2}}$$
$$\frac{\left(a^{-3}+b^{-3}\right)a^{3}b^{3}+\left(-a-b\right)\left(a^{2}-ab+b^{2}\right)}{a^{2}-ab+b^{2}}$$
$$\frac{b^{3}+a^{3}-a^{3}+a^{2}b-ab^{2}-ba^{2}+b^{2}a-b^{3}}{a^{2}-ab+b^{2}}$$
$$\frac{0}{a^{2}-ab+b^{2}}$$
$$0$$
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