Phân tích các đa thức sau thành nhân tử:
a. \(45 + {x^3} - 5{x^2} - 9x\)
b. \({x^4} - 2{x^3} - 2{x^2} - 2x - 3\)
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a. \(45 + {x^3} - 5{x^2} - 9x\) \( = \left( {{x^3} - 5{x^2}} \right) - \left( {9x - 45} \right) = {x^2}\left( {x - 5} \right) - 9\left( {x - 5} \right)\)
\( = \left( {x - 5} \right)\left( {{x^2} - 9} \right) = \left( {x - 5} \right)\left( {x - 3} \right)\left( {x + 3} \right)\)
b. \({x^4} - 2{x^3} - 2{x^2} - 2x - 3 = \left( {{x^4} - 1} \right) - \left( {2{x^3} + 2{x^2}} \right) - \left( {2x + 2} \right)\)
\(\eqalign{ & = \left( {{x^2} + 1} \right)\left( {{x^2} - 1} \right) - 2{x^2}\left( {x + 1} \right) - 2\left( {x + 1} \right) \cr & = \left( {{x^2} + 1} \right)\left( {x - 1} \right)\left( {x + 1} \right) - 2{x^2}\left( {x + 1} \right) - 2\left( {x + 1} \right) \cr & = \left( {x + 1} \right)\left[ {\left( {{x^2} + 1} \right)\left( {x - 1} \right) - 2{x^2} - 2} \right] \cr & = \left( {x + 1} \right)\left[ {\left( {{x^2} + 1} \right)\left( {x - 1} \right) - 2\left( {{x^2} + 1} \right)} \right] = \left( {x + 1} \right)\left( {{x^2} + 1} \right)\left( {x - 1 - 2} \right) \cr & = \left( {x + 1} \right)\left( {{x^2} + 1} \right)\left( {x - 3} \right) \cr} \)