$ \Rightarrow 2\overrightarrow a + 3\overrightarrow b = - \overrightarrow c $
Taking cross product with $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ respectively, we get
$2(\overrightarrow a \times \overrightarrow a ) + 3(\overrightarrow a \times \overrightarrow b ) = - \overrightarrow a \times \overrightarrow c $
$ \Rightarrow \quad 3(\overrightarrow a \times \overrightarrow b ) = \overrightarrow c \times \overrightarrow a $ .......$(i)$
and $\quad 2(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{a}})+3(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{b}})=-\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}$
$ \Rightarrow \quad 2(\overrightarrow a \times \overrightarrow b ) = \overrightarrow b \times \overrightarrow c $ .....$(ii)$
Now, $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}$
$ = \overrightarrow a \times \overrightarrow b + \overrightarrow b \times \overrightarrow c + 3(\overrightarrow a \times \overrightarrow b )\quad $ [using Eq. $(i)$]
$ = 4(\overrightarrow a \times \overrightarrow b ) + \overrightarrow b \times \overrightarrow c $
$ = 2(\overrightarrow b \times \overrightarrow c ) + \overrightarrow b \times \overrightarrow c $ [using Eq. $(ii)$]
$ = 3(\overrightarrow b \times \overrightarrow c )$
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