Explanation

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