= Solution
Use the <vector triple product identity> twice:
$$
(\mathbf a\times\mathbf b)\times\mathbf c=\mathbf b(\mathbf a\cdot\mathbf c)-\mathbf a(\mathbf b\cdot\mathbf c),\qquad \mathbf a\times(\mathbf b\times\mathbf c)=\mathbf b(\mathbf a\cdot\mathbf c)-\mathbf c(\mathbf a\cdot\mathbf b).
$$
Subtracting cancels the common term, yielding
$$
\boxed{(\mathbf a\times\mathbf b)\times\mathbf c-\mathbf a\times(\mathbf b\times\mathbf c)=\mathbf c(\mathbf a\cdot\mathbf b)-\mathbf a(\mathbf b\cdot\mathbf c).}
$$
It is generally nonzero: the <cross product> does not satisfy the <associative property>.
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