Đáp án:

\(\begin{array}{l}
\left( {x - y - 1} \right)\left( {x + y + 1} \right)\\
3.\left( {x - y - 2} \right)\left( {x - y + 2} \right)\\
\left( {x + y} \right)\left( {3 - x - y} \right)\\
\left( {x + y} \right)\left( {x - y - 2} \right)\\
\left( {x + y} \right)\left( {x + y - 1} \right)\left( {x + y + 1} \right)\\
18.\left( {m - n - 2p} \right)\left( {m - n + 2p} \right)
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
*)\\
{x^2} - \left( {{y^2} + 2y + 1} \right)\\
 = {x^2} - \left( {{y^2} + 2.y.1 + {1^2}} \right)\\
 = {x^2} - {\left( {y + 1} \right)^2}\\
 = \left[ {x - \left( {y + 1} \right)} \right].\left[ {x + \left( {y + 1} \right)} \right]\\
 = \left( {x - y - 1} \right)\left( {x + y + 1} \right)\\
*)\\
3{x^2} + 3{y^2} - 6xy - 12\\
 = 3.\left( {{x^2} + {y^2} - 2xy - 4} \right)\\
 = 3.\left[ {\left( {{x^2} - 2xy + {y^2}} \right) - 4} \right]\\
 = 3.\left[ {{{\left( {x - y} \right)}^2} - {2^2}} \right]\\
 = 3.\left[ {\left( {x - y} \right) - 2} \right].\left[ {\left( {x - y} \right) + 2} \right]\\
 = 3.\left( {x - y - 2} \right)\left( {x - y + 2} \right)\\
*)\\
3x + 3y - {x^2} - 2xy - {y^2}\\
 = \left( {3x + 3y} \right) - \left( {{x^2} + 2xy + {y^2}} \right)\\
 = 3.\left( {x + y} \right) - {\left( {x + y} \right)^2}\\
 = \left( {x + y} \right).\left[ {3 - \left( {x + y} \right)} \right]\\
 = \left( {x + y} \right)\left( {3 - x - y} \right)\\
*)\\
{x^2} - {y^2} - 2x - 2y\\
 = \left( {{x^2} - {y^2}} \right) - \left( {2x + 2y} \right)\\
 = \left( {x - y} \right)\left( {x + y} \right) - 2.\left( {x + y} \right)\\
 = \left( {x + y} \right).\left[ {\left( {x - y} \right) - 2} \right]\\
 = \left( {x + y} \right)\left( {x - y - 2} \right)\\
*)\\
{x^3} - x + 3{x^2}y + 3x{y^2} - y + {y^3}\\
 = \left( {{x^3} + 3{x^2}y + 3x{y^2} + {y^3}} \right) + \left( { - x - y} \right)\\
 = {\left( {x + y} \right)^3} - \left( {x + y} \right)\\
 = \left( {x + y} \right).\left[ {{{\left( {x + y} \right)}^2} - 1} \right]\\
 = \left( {x + y} \right).\left[ {{{\left( {x + y} \right)}^2} - {1^2}} \right]\\
 = \left( {x + y} \right).\left[ {\left( {x + y} \right) - 1} \right].\left[ {\left( {x + y} \right) + 1} \right]\\
 = \left( {x + y} \right)\left( {x + y - 1} \right)\left( {x + y + 1} \right)\\
*)\\
18{m^2} - 36mn + 18{n^2} - 72{p^2}\\
 = 18\left( {{m^2} - 2mn + {n^2} - 4{p^2}} \right)\\
 = 18.\left[ {\left( {{m^2} - 2mn + {n^2}} \right) - 4{p^2}} \right]\\
 = 18.\left[ {{{\left( {m - n} \right)}^2} - {{\left( {2p} \right)}^2}} \right]\\
 = 18.\left[ {\left( {m - n} \right) - 2p} \right].\left[ {\left( {m - n} \right) + 2p} \right]\\
 = 18.\left( {m - n - 2p} \right)\left( {m - n + 2p} \right)
\end{array}\)