\(\begin{array}{l}
\,\,\,\,\,{\left( {a + b + c} \right)^3} - {a^3} - {b^3} - {c^3} = 3\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)\\
VT = {\left( {a + b} \right)^3} + {c^3} + 3\left( {a + b} \right)c\left( {a + b + c} \right) - {a^3} - {b^3} - {c^3}\\
VT = {\left( {a + b} \right)^3} + 3\left( {a + b} \right)c\left( {a + b + c} \right) - {a^3} - {b^3}\\
VT = {a^3} + {b^3} + 3ab\left( {a + b} \right) + 3\left( {a + b} \right)c\left( {a + b + c} \right) - {a^3} - {b^3}\\
VT = 3ab\left( {a + b} \right) + 3\left( {a + b} \right)c\left( {a + b + c} \right)\\
VT = 3\left( {a + b} \right)\left[ {ab + c\left( {a + b + c} \right)} \right]\\
VT = 3\left( {a + b} \right)\left( {ab + ac + bc + {c^2}} \right)\\
VT = 3\left( {a + b} \right)\left[ {a\left( {b + c} \right) + c\left( {b + c} \right)} \right]\\
VT = 3\left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right)
\end{array}\)
