Simplify :
$(6 m - n) (36 m^{2} + 6 m n + n^{2}) - (3 m + 2 n)^{3}$

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Solution

$(6 m - n) (36 m^{2} + 6 m n + n^{2}) - (3 m + 2 n)^{2}$
$= (6 m - n) [(6 m)^{2} + 6 m \times n + (n)^{2}] - (3 m + 2 n)^{3}$
$= [(6 m)^{3} - n^{3}] - [(3 m)^{3} + (2 n)^{3} + 3 \times 3 m \times 2 n (3 m + 2 n)]$
$= (216 m^{3} - n^{3}) - (27 m^{3} + 8 n^{3} + 54 m^{2} n + 36 m n^{2})$
$= 216 m^{3} - n^{3} - 27 m^{3} - 8 n^{3} - 54 m^{2} n - 36 m n^{2}$
$= 189 m^{3} - 9 n^{3} - 54 m^{2} n - 36 m n^{2}$

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