Factorise the following: $(2 m + 3 n)^{3} - (3 m + 2 n)^{3}$

(- m + n) (19 m^{2} + 37 m n + 19 n^{2})

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(m + 2 n) (19 m^{2} + 37 m n - 19 n^{2})

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(- m + n) (19 m^{2} - 37 m n + 19 n^{2})

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(m + n) (19 m^{2} + 37 m n - 19 n^{2})

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Solution

The correct option is A $(- m + n) (19 m^{2} + 37 m n + 19 n^{2})$
$(2 m + 3 n)^{3} - (3 m + 2 n)^{3}$
Using, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$= [(2 m + 3 n) - (3 m + 2 n)] [(2 m + 3 n)^{2} + (2 m + 3 n) (3 m + 2 n) + (3 m + 2 n)^{2}]$
$= [2 m + 3 n - 3 m - 2 n] [4 m^{2} + 12 m n + 9 n^{2} + 6 m^{2} + 4 m n + 9 m n + 6 n^{2} + 9 m^{2} + 12 m n + 4 n^{2}]$
$= (- m + n) (19 m^{2} + 37 m n + 19 n^{2})$

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