Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
3,\\
a,\\
\left( {5{x^2} + 9xy - 2{y^2}} \right):\left( {x + 2y} \right)\\
= \left[ {\left( {5{x^2} + 10xy} \right) + \left( { - xy - 2{y^2}} \right)} \right]:\left( {x + 2y} \right)\\
= \left[ {5x.\left( {x + 2y} \right) - y.\left( {x + 2y} \right)} \right]:\left( {x + 2y} \right)\\
= \left[ {\left( {x + 2y} \right)\left( {5x - y} \right)} \right]:\left( {x + 2y} \right)\\
= 5x - y\\
b,\\
\left( {{x^4} - {x^3}y + {x^2}{y^2} - x{y^3}} \right):\left( {{x^2} + {y^2}} \right)\\
= \left[ {\left( {{x^4} - {x^3}y} \right) + \left( {{x^2}{y^2} - x{y^3}} \right)} \right]:\left( {{x^2} + {y^2}} \right)\\
= \left[ {{x^3}\left( {x - y} \right) + x{y^2}\left( {x - y} \right)} \right]:\left( {{x^2} + {y^2}} \right)\\
= \left[ {\left( {x - y} \right)\left( {{x^3} + x{y^2}} \right)} \right]:\left( {{x^2} + {y^2}} \right)\\
= \left[ {\left( {x - y} \right).x.\left( {{x^2} + {y^2}} \right)} \right]:\left( {{x^2} + {y^2}} \right)\\
= x.\left( {x - y} \right)\\
c,\\
\left( {4{x^5} + 3x{y^4} - {y^5} + 2{x^4}y - 6{x^3}{y^2}} \right):\left( {2{x^3} + {y^3} - 2x{y^2}} \right)\\
= \left[ {\left( {4{x^5} + 2{x^2}{y^3} - 4{x^3}{y^2}} \right) + \left( { - 2{x^3}{y^2} - {y^5} + 2x{y^4}} \right) + \left( {x{y^4} + 2{x^4}y - 2{x^2}{y^3}} \right)} \right]:\left( {2{x^3} + {y^3} - 2x{y^2}} \right)\\
= \left[ {2{x^2}.\left( {2{x^3} + {y^3} - 2x{y^2}} \right) - {y^2}\left( {2{x^3} + {y^3} - 2x{y^2}} \right) + xy.\left( {{y^3} + 2{x^3} - 2x{y^2}} \right)} \right]:\left( {2{x^3} + {y^3} - 2x{y^2}} \right)\\
= \left[ {\left( {2{x^3} + {y^3} - 2x{y^2}} \right)\left( {2{x^2} - {y^2} + xy} \right)} \right]:\left( {2{x^3} + {y^3} - 2x{y^2}} \right)\\
= 2{x^2} - {y^2} + xy\\
d,\\
\left( {2{a^3} + 7a{b^2} - 7a{b^2} - 2{b^3}} \right):\left( {2a - b} \right)\\
= \left[ {\left( {2{a^3} - 2{b^3}} \right) + \left( {7a{b^2} - 7a{b^2}} \right)} \right]:\left( {2a - b} \right)\\
= \left[ {2.\left( {{a^3} - {b^3}} \right) + 7ab.\left( {b - a} \right)} \right]:\left( {2a - b} \right)\\
= \left[ {2.\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) - 7ab\left( {a - b} \right)} \right]:\left( {2a - b} \right)\\
= \left[ {\left( {a - b} \right).\left( {2{a^2} + 2ab + 2{b^2} - 7ab} \right)} \right]:\left( {2a - b} \right)\\
= \left[ {\left( {a - b} \right)\left( {2{a^2} - 5ab + 2{b^2}} \right)} \right]:\left( {2a - b} \right)\\
= \left[ {\left( {a - b} \right).\left[ {\left( {2{a^2} - 4ab} \right) + \left( { - ab + 2{b^2}} \right)} \right]} \right]:\left( {2a - b} \right)\\
= \left[ {\left( {a - b} \right).\left[ {2a.\left( {a - 2b} \right) - b.\left( {a - 2b} \right)} \right]} \right]:\left( {2a - b} \right)\\
= \left[ {\left( {a - b} \right).\left( {a - 2b} \right).\left( {2a - b} \right)} \right]:\left( {2a - b} \right)\\
= \left( {a - b} \right).\left( {a - 2b} \right)\\
4,\\
{\left( {2x + 4y} \right)^2}:\left( {x + 2y} \right) - \left( {9{x^3} - 12{x^2} - 3x} \right):\left( { - 3x} \right) - 3.\left( {{x^2} + 3} \right)\\
= {\left[ {2.\left( {x + 2y} \right)} \right]^2}:\left( {x + 2y} \right) - \left[ {3x.3{x^2} - 3x.4x - 3x} \right]:\left( { - 3x} \right) - 3{x^2} - 9\\
= 4.{\left( {x + 2y} \right)^2}:\left( {x + 2y} \right) - \left[ {3x.\left( {3{x^2} - 4x - 1} \right)} \right]:\left( { - 3x} \right) - 3{x^2} - 9\\
= 4.\left( {x + 2y} \right) + \left( {3{x^2} - 4x - 1} \right) - 3{x^2} - 9\\
= 8y - 10
\end{array}\)
