Đáp án:

\(\begin{array}{l}
3,\\
a,\,\,\,4xy\\
b,\,\,\,4.x.\left( {2x + 1} \right)\\
c,\,\,\,\left( {x + y + z} \right).\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)\\
4,\\
a,\,\,\,400\\
b,\,\,\,12000
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
3,\\
a,\,\,\,{\left( {x + y} \right)^2} - {\left( {x - y} \right)^2}\\
 = \left[ {\left( {x + y} \right) - \left( {x - y} \right)} \right].\left[ {\left( {x + y} \right) + \left( {x - y} \right)} \right]\\
 = \left( {x + y - x + y} \right).\left( {x + y + x - y} \right)\\
 = 2y.2x\\
 = 4xy\\
b,\,\,\,{\left( {3x + 1} \right)^2} - {\left( {x + 1} \right)^2}\\
 = \left[ {\left( {3x + 1} \right) - \left( {x + 1} \right)} \right].\left[ {\left( {3x + 1} \right) + \left( {x + 1} \right)} \right]\\
 = \left( {3x + 1 - x - 1} \right).\left( {3x + 1 + x + 1} \right)\\
 = 2x.\left( {4x + 2} \right)\\
 = 2x.2.\left( {2x + 1} \right)\\
 = 4.x.\left( {2x + 1} \right)\\
c,\,\,\,{x^3} + {y^3} + {z^3} - 3xyz\\
 = \left( {{x^3} + 3{x^2}y + 3x{y^2} + {y^3}} \right) + {z^3} - 3{x^2}y - 3x{y^2} - 3xyz\\
 = {\left( {x + y} \right)^3} + {z^3} - 3xy\left( {x + y + z} \right)\\
 = \left[ {\left( {x + y} \right) + z} \right].\left[ {{{\left( {x + y} \right)}^2} - \left( {x + y} \right).z + {z^2}} \right] - 3xy\left( {x + y + z} \right)\\
 = \left( {x + y + z} \right).\left( {{x^2} + 2xy + {y^2} - xz - yz + {z^2}} \right) - 3xy\left( {x + y + z} \right)\\
 = \left( {x + y + z} \right).\left[ {\left( {{x^2} + 2xy + {y^2} - xz - yz + {z^2}} \right) - 3xy} \right]\\
 = \left( {x + y + z} \right).\left( {{x^2} + {y^2} + {z^2} - xy - yz - zx} \right)\\
4,\\
a,\,\,\,{25^2} - {15^2} = \left( {25 - 15} \right)\left( {25 + 15} \right) = 10.40 = 400\\
b,\,\,\,{87^2} + {73^2} - {27^2} - {13^2}\\
 = \left( {{{87}^2} - {{13}^2}} \right) + \left( {{{73}^2} - {{27}^2}} \right)\\
 = \left( {87 - 13} \right).\left( {87 + 13} \right) + \left( {73 - 27} \right).\left( {73 + 27} \right)\\
 = 74.100 + 46.100\\
 = 100.\left( {74 + 46} \right)\\
 = 100.120\\
 = 12000
\end{array}\)