Step-by-step Solution:

\[ (\vec a + \vec b + \vec c) \cdot \left[ \left( \vec a + \vec b \right) \times \left( \vec a + \vec c \right) \right] \] This can be expanded as: \[ (\vec a + \vec b + \vec c) \cdot \left[ \vec a \times \vec a + \vec a \times \vec c + \vec b \times \vec a + \vec b \times \vec c \right] \] This simplifies to: \[ \vec a \cdot (\vec b \times \vec c) + \vec b \cdot (\vec a \times \vec c) + \vec c \cdot (\vec b \times \vec a) \] And further: \[ \left[ \vec a \, \vec b \, \vec c \right] + \left[ \vec b \, \vec a \, \vec c \right] + \left[ \vec c \, \vec b \, \vec a \right] = - \left[ \vec a \, \vec b \, \vec c \right] \]