Giải thích các bước giải:
$\begin{array}{l}
a({b^2} + {c^2}) + b({c^2} + {a^2}) + c({a^2} + {b^2}) - 2abc - {a^3} - {b^3} - {c^3}\\
= a\left( {{b^2} + {c^2} - bc} \right) + b({c^2} + {a^2}) + c({a^2} + {b^2}) - abc - {a^3} - {b^3} - {c^3}\\
= \left( {a\left( {{b^2} + {c^2} - bc} \right) - \left( {{b^3} + {c^3}} \right)} \right) + b({c^2} + {a^2}) + c({a^2} + {b^2}) - abc - {a^3}\\
= \left( {a\left( {{b^2} + {c^2} - bc} \right) - \left( {b + c} \right)\left( {{b^2} + {c^2} - bc} \right)} \right) + \left( {b{c^2} + {b^2}c - abc} \right) + \left( {{a^2}b + {a^2}c - {a^3}} \right)\\
= \left( {{b^2} + {c^2} - bc} \right)\left( {a - b - c} \right) + bc\left( {b + c - a} \right) + {a^2}\left( {b + c - a} \right)\\
= \left( {a - b - c} \right)\left( {{b^2} + {c^2} - bc - bc - {a^2}} \right)\\
= \left( {a - b - c} \right)\left( {{b^2} - 2bc + {c^2} - {a^2}} \right)\\
= \left( {a - b - c} \right)\left( {{{\left( {b - c} \right)}^2} - {a^2}} \right)\\
= \left( {a - b - c} \right)\left( {b - c - a} \right)\left( {b - c + a} \right)
\end{array}$
