A) \[\left( a+\frac{2}{a} \right)\left( {{a}^{2}}+\frac{4}{{{a}^{2}}}-2 \right)\]
B) \[\left( a+\frac{2}{a} \right)\left( {{a}^{2}}+\frac{4}{{{a}^{2}}}+2 \right)\]
C) \[\left( a-\frac{2}{a} \right)\left( {{a}^{2}}+\frac{4}{{{a}^{2}}}+2 \right)\]
D) \[\left( a-\frac{2}{a} \right)\left( {{a}^{2}}+\frac{4}{{{a}^{2}}}-2 \right)\]
Correct Answer: D
Solution :
\[{{a}^{3}}-\frac{8}{{{a}^{3}}}-4a+\frac{8}{a}={{a}^{3}}-\frac{8}{{{a}^{3}}}-4\left( a-\frac{2}{a} \right)\] \[=\left[ {{(a)}^{3}}-{{\left( \frac{2}{a} \right)}^{3}} \right]-4\left( a-\frac{2}{a} \right)\] \[=\left( a-\frac{2}{a} \right)\left( {{a}^{2}}+2+\frac{4}{{{a}^{2}}} \right)-4\left( a-\frac{2}{a} \right)\] \[=\left( a-\frac{2}{a} \right)\left( {{a}^{2}}+\frac{4}{{{a}^{2}}}-4+2 \right)=\left( a-\frac{2}{a} \right)\left( {{a}^{2}}+\frac{4}{{{a}^{2}}}-2 \right)\]
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