Explanation

(a)
$\overrightarrow{\mathbf{a}} \bullet \left\lbrack \left( \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) \times \left( \ \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} \right) \right\rbrack$
$= \overrightarrow{\mathbf{a}} \bullet \lbrack\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{b}})\rbrack$
$= \overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) + \overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{b}} \right)$
$= \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack + \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{b}} \right\rbrack = 0\ \ \ \ \ \ \ \left\lbrack \because\left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{b}} \right\rbrack = - \left\lbrack \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}} \right\rbrack \right\rbrack$