Đáp án:

\(\begin{array}{l}
a,\\
3.\left( {x + 1} \right)\left( {3x - 5} \right)\\
b,\\
 - 8\left( {x + 2} \right)\left( {3x - 1} \right)\\
c,\\
\left( {x + 14} \right).\left( {3x - 4} \right)\\
d,\\
\left( {x + 5} \right).\left( {5x - 3} \right)\\
e,\\
\left( {4x + 7} \right).\left( {8x + 11} \right)\\
f,\\
\left( {a - b + c} \right).\left( {a + b - c} \right).\left( { - a + b + c} \right).\left( {a + b + c} \right)
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
a,\\
{\left( {3x - 1} \right)^2} - 16\\
 = {\left( {3x - 1} \right)^2} - {4^2}\\
 = \left[ {\left( {3x - 1} \right) - 4} \right].\left[ {\left( {3x - 1} \right) + 4} \right]\\
 = \left( {3x - 1 - 4} \right)\left( {3x - 1 + 4} \right)\\
 = \left( {3x - 5} \right)\left( {3x + 3} \right)\\
 = \left( {3x - 5} \right).3.\left( {x + 1} \right)\\
 = 3.\left( {x + 1} \right)\left( {3x - 5} \right)\\
b,\\
{\left( {5x - 4} \right)^2} - 49{x^2}\\
 = {\left( {5x - 4} \right)^2} - {\left( {7x} \right)^2}\\
 = \left[ {\left( {5x - 4} \right) - 7x} \right].\left[ {\left( {5x - 4} \right) + 7x} \right]\\
 = \left( {5x - 4 - 7x} \right)\left( {5x - 4 + 7x} \right)\\
 = \left( { - 2x - 4} \right).\left( {12x - 4} \right)\\
 = \left( { - 2} \right).\left( {x + 2} \right).4.\left( {3x - 1} \right)\\
 =  - 8\left( {x + 2} \right)\left( {3x - 1} \right)\\
c,\\
{\left( {2x + 5} \right)^2} - {\left( {x - 9} \right)^2}\\
 = \left[ {\left( {2x + 5} \right) - \left( {x - 9} \right)} \right].\left[ {\left( {2x + 5} \right) + \left( {x - 9} \right)} \right]\\
 = \left( {2x + 5 - x + 9} \right).\left( {2x + 5 + x - 9} \right)\\
 = \left( {x + 14} \right).\left( {3x - 4} \right)\\
d,\\
{\left( {3x + 1} \right)^2} - 4.{\left( {x - 2} \right)^2}\\
 = {\left( {3x + 1} \right)^2} - {2^2}.{\left( {x - 2} \right)^2}\\
 = {\left( {3x + 1} \right)^2} - {\left[ {2.\left( {x - 2} \right)} \right]^2}\\
 = {\left( {3x + 1} \right)^2} - {\left( {2x - 4} \right)^2}\\
 = \left[ {\left( {3x + 1} \right) - \left( {2x - 4} \right)} \right].\left[ {\left( {3x + 1} \right) + \left( {2x - 4} \right)} \right]\\
 = \left( {3x + 1 - 2x + 4} \right).\left( {3x + 1 + 2x - 4} \right)\\
 = \left( {x + 5} \right).\left( {5x - 3} \right)\\
e,\\
9{\left( {2x + 3} \right)^2} - 4.{\left( {x + 1} \right)^2}\\
 = {3^2}.{\left( {2x + 3} \right)^2} - {2^2}.{\left( {x + 1} \right)^2}\\
 = {\left[ {3.\left( {2x + 3} \right)} \right]^2} - {\left[ {2.\left( {x + 1} \right)} \right]^2}\\
 = {\left( {6x + 9} \right)^2} - {\left( {2x + 2} \right)^2}\\
 = \left[ {\left( {6x + 9} \right) - \left( {2x + 2} \right)} \right].\left[ {\left( {6x + 9} \right) + \left( {2x + 2} \right)} \right]\\
 = \left( {6x + 9 - 2x - 2} \right).\left( {6x + 9 + 2x + 2} \right)\\
 = \left( {4x + 7} \right).\left( {8x + 11} \right)\\
f,\\
4{b^2}{c^2} - {\left( {{b^2} + {c^2} - {a^2}} \right)^2}\\
 = {\left( {2bc} \right)^2} - {\left( {{b^2} + {c^2} - {a^2}} \right)^2}\\
 = \left[ {2bc - \left( {{b^2} + {c^2} - {a^2}} \right)} \right].\left[ {2bc + \left( {{b^2} + {c^2} - {a^2}} \right)} \right]\\
 = \left( {2bc - {b^2} - {c^2} + {a^2}} \right).\left( {2bc + {b^2} + {c^2} - {a^2}} \right)\\
 = \left[ {{a^2} - \left( {{b^2} - 2bc + {c^2}} \right)} \right].\left[ {\left( {{b^2} + 2bc + {c^2}} \right) - {a^2}} \right]\\
 = \left[ {{a^2} - {{\left( {b - c} \right)}^2}} \right].\left[ {{{\left( {b + c} \right)}^2} - {a^2}} \right]\\
 = \left[ {a - \left( {b - c} \right)} \right].\left[ {a + \left( {b - c} \right)} \right].\left[ {\left( {b + c} \right) - a} \right].\left[ {\left( {b + c} \right) + a} \right]\\
 = \left( {a - b + c} \right).\left( {a + b - c} \right).\left( { - a + b + c} \right).\left( {a + b + c} \right)
\end{array}\)