Giải thích các bước giải:
Ta có:
$\begin{array}{l}
{x^2}{y^2}.(y - x) + {y^2}{z^2}.(z - y) - {z^2}{x^2}.(z - x)\\
= \left( {{x^2}{y^2}.(y - x) - {z^2}{x^2}.(z - x)} \right) + {y^2}{z^2}.(z - y)\\
= {x^2}\left( {{y^3} - x{y^2} - {z^3} + x{z^2}} \right) + {y^2}{z^2}.(z - y)\\
= {x^2}\left( {\left( {{y^3} - {z^3}} \right) - x\left( {{y^2} - {z^2}} \right)} \right) + {y^2}{z^2}.(z - y)\\
= {x^2}\left( {y - z} \right)\left( {{y^2} + yz + {z^2} - xy - xz} \right) - {y^2}{z^2}\left( {y - z} \right)\\
= \left( {y - z} \right)\left( {{x^2}\left( {{y^2} + yz + {z^2} - xy - xz} \right) - {y^2}{z^2}} \right)\\
= \left( {y - z} \right)\left( {{x^2}{y^2} + {x^2}yz + {x^2}{z^2} - {x^3}y - {x^3}z - {y^2}{z^2}} \right)\\
= \left( {y - z} \right)\left( {\left( {{x^2}{y^2} - {y^2}{z^2}} \right) + \left( {{x^2}yz - {x^3}y} \right) + \left( {{x^2}{z^2} - {x^3}z} \right)} \right)\\
= \left( {y - z} \right)\left( {{y^2}\left( {{x^2} - {z^2}} \right) + {x^2}y\left( {z - x} \right) + {x^2}z\left( {z - x} \right)} \right)\\
= \left( {y - z} \right)\left( {z - x} \right)\left( { - {y^2}x - {y^2}z + {x^2}y + {x^2}z} \right)\\
= \left( {y - z} \right)\left( {z - x} \right)\left( {\left( {{x^2}y - {y^2}x} \right) + \left( {{x^2}z - {y^2}z} \right)} \right)\\
= \left( {y - z} \right)\left( {z - x} \right)\left( {xy\left( {x - y} \right) + z\left( {{x^2} - {y^2}} \right)} \right)\\
= \left( {y - z} \right)\left( {z - x} \right)\left( {x - y} \right)\left( {xy + xz + yz} \right)
\end{array}$
