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Find the product of $(a^{2} - b^{2}) and (2 a + b) .$

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$2 a^{3} + a^{2} b + 2 a b^{2} + b^{3}$

$2 a^{3} + a^{2} b - 2 a b^{2} - b^{3}$

$2 a^{3} - a^{2} b - 2 a b^{2} - b^{3}$

$2 a^{3} - a^{2} b + 2 a b^{2} - b^{3}$

answer is B.

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Detailed Solution

$(a^{2} - b^{2}) (2 a + b)$

$= a^{2} (2 a + b) - b^{2} (2 a + b)$

$= a^{2} (2 a) + a^{2} (b) - b^{2} (2 a) - b^{2} (b)$

$= 2 a^{3} + a^{2} b - 2 a b^{2} - b^{3}$

Hence, the product of $(a^{2} - b^{2}) and (2 a + b)$ is $2 a^{3} + a^{2} b - 2 a b^{2} - b^{3} .$

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