Giải thích các bước giải:

\(\begin{array}{l}
1,\\
\left( {a + b + c} \right)\left( {ab + bc + ca} \right) - abc\\
 = \left( {{a^2}b + abc + {a^2}c + a{b^2} + {b^2}c + abc + abc + b{c^2} + {c^2}a} \right) - abc\\
 = {a^2}b + a{b^2} + {a^2}c + a{c^2} + {b^2}c + b{c^2} + 2abc\\
 = \left( {{a^2}b + a{b^2}} \right) + \left( {{a^2}c + 2abc + {b^2}c} \right) + \left( {a{c^2} + b{c^2}} \right)\\
 = ab\left( {a + b} \right) + c.\left( {{a^2} + 2ab + {b^2}} \right) + {c^2}\left( {a + b} \right)\\
 = ab\left( {a + b} \right) + c.{\left( {a + b} \right)^2} + {c^2}\left( {a + b} \right)\\
 = \left( {a + b} \right).\left( {ab + c.\left( {a + b} \right) + {c^2}} \right)\\
 = \left( {a + b} \right).\left( {ab + ca + cb + {c^2}} \right)\\
 = \left( {a + b} \right).\left( {a\left( {b + c} \right) + c\left( {b + c} \right)} \right)\\
 = \left( {a + b} \right).\left( {b + c} \right).\left( {c + a} \right)\\
2,\\
4a\left( {a + b} \right)\left( {a + b + c} \right).\left( {a + c} \right) + {b^2}{c^2}\\
 = 4.\left[ {a.\left( {a + b + c} \right)} \right].\left[ {\left( {a + b} \right)\left( {a + c} \right)} \right] + {b^2}{c^2}\\
 = 4.\left( {{a^2} + ab + ac} \right).\left( {{a^2} + ab + ac + bc} \right) + {b^2}{c^2}\\
 = 4.{\left( {{a^2} + ab + ac} \right)^2} + 4bc.\left( {{a^2} + ab + ac} \right) + {b^2}{c^2}\\
 = {\left[ {2.\left( {{a^2} + ab + ac} \right)} \right]^2} + 2.2.\left( {{a^2} + ab + ac} \right).bc + {\left( {bc} \right)^2}\\
 = {\left( {2{a^2} + 2ab + 2ac + bc} \right)^2}\\
3,\\
a\left( {{b^2} + {c^2}} \right) + b\left( {{c^2} + {a^2}} \right) + c\left( {{a^2} + {b^2}} \right) + 3abc\\
 = a{b^2} + a{c^2} + b{c^2} + b{a^2} + c{a^2} + c{b^2} + 3abc\\
 = \left( {a{b^2} + {a^2}b + abc} \right) + \left( {a{c^2} + c{a^2} + abc} \right) + \left( {b{c^2} + c{b^2} + abc} \right)\\
 = ab\left( {a + b + c} \right) + ac\left( {a + b + c} \right) + bc\left( {a + b + c} \right)\\
 = \left( {a + b + c} \right).\left( {ab + ac + bc} \right)\\
4,\\
ab\left( {a + b} \right) - bc\left( {b + c} \right) - ca\left( {c - a} \right)\\
 = ab\left( {a + b} \right) - {b^2}c - b{c^2} - {c^2}a + c{a^2}\\
 = ab\left( {a + b} \right) + \left( {c{a^2} - {b^2}c} \right) - \left( {b{c^2} + {c^2}a} \right)\\
 = ab\left( {a + b} \right) + c.\left( {{a^2} - {b^2}} \right) - {c^2}\left( {a + b} \right)\\
 = ab\left( {a + b} \right) + c.\left( {a - b} \right)\left( {a + b} \right) - {c^2}\left( {a + b} \right)\\
 = \left( {a + b} \right).\left( {ab + ca - cb - {c^2}} \right)\\
 = \left( {a + b} \right).\left( {a.\left( {b + c} \right) - c\left( {b + c} \right)} \right)\\
 = \left( {a + b} \right).\left( {b + c} \right)\left( {a - c} \right)
\end{array}\)