In the expansion of $(2 x^{2} - 8) {(x - 4)}^{2}$ ; find
Coefficient of $x^{3}$ .

- 26

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9

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1

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- 16

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Solution

$(2 x^{2} - 8) {(x - 4)}^{2}$
$= (2 x^{2} - 8) (x^{2} - 2 x (4) + 4^{2})$ [Since, $(a - b)^{2} = a^{2} - 2 a b + b^{2}$ ]
$= (2 x^{2} - 8) (x^{2} - 8 x + 16)$
$= (2 x^{2} \times x^{2}) - (2 x^{2} \times 8 x) + (2 x^{2} \times 16) - (8 \times x^{2}) + (8 \times 8 x) - (8 \times 16)$
$= 2 x^{4} - 16 x^{3} + 32 x^{2} - 8 x^{2} + 64 x - 128$ [Since, $a^{m} \times a^{n} = a^{m + n}$ ]
$= 2 x^{4} - 16 x^{3} + 24 x^{2} + 64 x - 128$
$\Rightarrow$ Coefficient of $x^{3}$ is $- 16$
Hence, Option D is correct.

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