\[\begin{array}{l}
{(a + b + c)^3} = {a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)\\
Xet:VT = \\
{(a + b + c)^3}\\
= {\left( {a + b} \right)^3} + 3{\left( {a + b} \right)^2}c + 3\left( {a + b} \right){c^2} + {c^3}\\
= {a^3} + 3{{\rm{a}}^2}b + 3{\rm{a}}{b^2} + {b^3} + 3\left( {{a^2} + 2{\rm{a}}b + {b^2}} \right)c + 3a{c^2} + 3b{c^2} + {c^3}\\
= {a^3} + {b^3} + {c^3} + 3{{\rm{a}}^2}b + 3{\rm{a}}{b^2} + 3{{\rm{a}}^2}c + 6{\rm{a}}bc + 3{b^2}c + 3{\rm{a}}{c^2} + 3b{c^2}\\
= {a^3} + {b^3} + {c^3} + 3{\rm{a}}b\left( {a + b} \right) + 3{{\rm{a}}^2}c + 3{\rm{a}}bc + 3{\rm{a}}bc + 3{b^2}c + 3{\rm{a}}{c^2} + 3b{c^2}\\
= {a^3} + {b^3} + {c^3} + 3{\rm{a}}b\left( {a + b} \right) + 3{\rm{a}}c\left( {a + b} \right) + 3bc\left( {a + b} \right) + 3{c^2}\left( {a + b} \right)\\
= {a^3} + {b^3} + {c^3} + 3{\rm{a}}b\left( {a + b} \right) + 3\left( {a + b} \right)\left( {ac + bc + {c^2}} \right)\\
= {a^3} + {b^3} + {c^3} + 3{\rm{a}}b\left( {a + b} \right)\left( {ab + ac + bc + {c^2}} \right)\\
= {a^3} + {b^3} + {c^3} + 3{\rm{a}}b\left( {a + b} \right)\left[ {a\left( {b + c} \right) + c\left( {b + c} \right)} \right]\\
= {a^3} + {b^3} + {c^3} + 3{\rm{a}}b\left( {a + b} \right)\left( {b + c} \right)\left( {a + c} \right) = VP\\
vay:{(a + b + c)^3} = {a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)
\end{array}\]