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Rút gọn các biểu thức sau:
a) \((x + 3)(x – 2) + {(x – 3)^2}\) ;
b) \({(x + 2)^2} + (x + 1)(x – 1)\) ;
c) \((2x – 1)(2x + 1) – {(2x – 3)^2}\) ;
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d) \({(x + 2)^3} – {(x + 1)^3} – {(x + 1)^2}\).
\(\eqalign{ & a)\,\,\left( {x + 3} \right)\left( {x – 2} \right) + {\left( {x – 3} \right)^2} \cr & \,\,\,\,\, = {x^2} – 2x + 3x – 6 + {x^2} – 6x + 9 \cr & \,\,\,\,\, = 2{x^2} – 5x + 3 \cr & b)\,\,{\left( {x + 2} \right)^2} + \left( {x + 1} \right)\left( {x – 1} \right) \cr & \,\,\,\,\, = {x^2} + 4x + 4 + {x^2} – 1 \cr & \,\,\,\,\, = 2{x^2} + 4x + 3 \cr & c)\,\,\left( {2x – 1} \right)\left( {2x + 1} \right) – {\left( {2x – 3} \right)^2} \cr & \,\,\,\,\, = {\left( {2x} \right)^2} – 1 – \left( {4{x^2} – 12x + 9} \right) \cr & \,\,\,\,\, = 4{x^2} – 1 – 4{x^2} + 12x – 9 \cr & \,\,\,\,\, = 12x – 10 \cr & d)\,\,{\left( {x + 2} \right)^3} – {\left( {x + 1} \right)^3} – {\left( {x + 1} \right)^2} \cr & \,\,\,\,\, = {x^3} + 6{x^2} + 12x + 8 – \left( {{x^3} + 3{x^2} + 3x + 1} \right) – \left( {{x^2} + 2x + 1} \right) \cr & \,\,\,\,\, = {x^3} + 6{x^2} + 12x + 8 – {x^3} – 3{x^2} – 3x – 1 – {x^2} – 2x – 1 \cr & \,\,\,\,\, = 2{x^2} + 7x + 6 \cr} \)