Giải thích các bước giải:

Ta có:

\(\begin{array}{l}
{x^2}{y^2}\left( {y - x} \right) + {y^2}{z^2}\left( {z - y} \right) - {x^2}{z^2}\left( {z - x} \right)\\
 = {x^2}{y^2}\left( {y - x} \right) + {y^2}{z^3} - {y^3}{z^2} - {x^2}{z^3} + {x^3}{z^2}\\
 = {x^2}{y^2}\left( {y - x} \right) + \left( {{y^2}{z^3} - {x^2}{z^3}} \right) + \left( {{x^3}{z^2} - {y^3}{z^2}} \right)\\
 = {x^2}{y^2}\left( {y - x} \right) + {z^3}\left( {{y^2} - {x^2}} \right) + {z^2}.\left( {{x^3} - {y^3}} \right)\\
 = {x^2}{y^2}\left( {y - x} \right) + {z^3}.\left( {y - x} \right).\left( {y + x} \right) + {z^2}.\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)\\
 = \left( {y - x} \right).\left( {{x^2}{y^2} + {z^3}.\left( {y + x} \right) - {z^2}.\left( {{x^2} + xy + {y^2}} \right)} \right)\\
 = \left( {y - x} \right).\left( {{x^2}{y^2} + {z^3}y + {z^3}x - {z^2}{x^2} - {z^2}xy - {z^2}{y^2}} \right)\\
 = \left( {y - x} \right).\left[ {\left( {{x^2}{y^2} - {z^2}{y^2}} \right) + \left( {{z^3}y - {z^2}xy} \right) + \left( {{z^3}x - {z^2}{x^2}} \right)} \right]\\
 = \left( {y - x} \right).\left[ {{y^2}\left( {{x^2} - {z^2}} \right) + {z^2}y\left( {z - x} \right) + {z^2}x\left( {z - x} \right)} \right]\\
 = \left( {y - x} \right).\left[ {{y^2}\left( {x - z} \right)\left( {x + z} \right) + {z^2}y\left( {z - x} \right) + {z^2}x.\left( {z - x} \right)} \right]\\
 = \left( {y - x} \right).\left( {x - z} \right).\left[ {{y^2}\left( {x + z} \right) - {z^2}y - {z^2}x} \right]\\
 = \left( {y - x} \right)\left( {x - z} \right).\left[ {\left( {{y^2}x - {z^2}x} \right) + \left( {{y^2}z - {z^2}y} \right)} \right]\\
 = \left( {y - x} \right).\left( {x - z} \right).\left[ {x\left( {{y^2} - {z^2}} \right) + zy\left( {y - z} \right)} \right]\\
 = \left( {y - x} \right).\left( {x - z} \right).\left[ {x\left( {y - z} \right)\left( {y + z} \right) + zy\left( {y - z} \right)} \right]\\
 = \left( {x - y} \right).\left( {z - x} \right)\left( {y - z} \right).\left[ {x\left( {y + z} \right) + zy} \right]\\
 = \left( {x - y} \right)\left( {y - z} \right)\left( {z - x} \right)\left( {xy + yz + zx} \right)
\end{array}\)