If $\vec a,\vec b,\vec c$ are three non-coplanar, non-zero vectors, then $(\vec a. \vec a) (\vec b×\vec c) + (\vec a.\vec b) (\vec c×\vec a) + (\vec a.\vec c) (\vec a×\vec b)$ is equal to

$[\vec b\,\,\vec c\,\,\vec a]\vec a$

Since $\vec a,\vec b,\vec c$ are non-coplanar vectors. Therefore, so are the vectors $\vec a×\vec b, \vec b×\vec c, \vec c×\vec a$.

We know that any vector in space is expressible as the linear combination of three non-coplanar vectors. So, let

$\vec a = x(\vec b×\vec c) + y (\vec c×\vec a) +z (\vec a×\vec b)$   ...(i)

Taking dot products successively with $\vec a,\vec b,\vec c$ we get

$x=\frac{\vec a.\vec a}{[\vec a\,\,\vec b\,\,\vec c]},y=\frac{\vec a.\vec b}{[\vec a\,\,\vec b\,\,\vec c]},z=\frac{\vec a.\vec c}{[\vec a\,\,\vec b\,\,\vec c]}$

Substituting these values in (i), we obtain that the given expression is equal to $[\vec a\,\,\vec b\,\,\vec c]\vec a$.