Factorise the following: $(a + b)^{3} - 8$

(a + b - 2) (a^{2} + 2 a b + b^{2} + 2 a + 2 b + 4)

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(a - b - 2) (a^{2} + 2 a b + b^{2} + 2 a - 2 b + 4)

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(a + b + 2) (a^{2} + 2 a b + b^{3} + 2 a + 2 b + 4)

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(a + b - 2) (b^{2} + 2 b + a^{2} + 2 a b + 2 b + 4)

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Solution

The correct option is A $(a + b - 2) (a^{2} + 2 a b + b^{2} + 2 a + 2 b + 4)$
$(a + b)^{3} - 8$
$= (a + b)^{3} - 2^{2}$
Using, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$= (a + b) - 2 [(a + b)^{2} + 2 (a + b) + (2^{2})]$
$= (a + b - 2) (a^{2} + 2 a b + b^{2} + 2 a + 2 b + 4)$

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