Find the following product:

$(i i i)$ $(2 a - 3 b - 2 c) (4 a^{2} + 9 b^{2} + 4 c^{2} + 6 a b - 6 b c + 4 c a)$

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Solution

Given,

$(2 a - 3 b - 2 c) (4 a^{2} + 9 b^{2} + 4 c^{2} + 6 a b - 6 b c + 4 c a)$

$= (2 a + (- 3 b) + (- 2 c)) ((2 a)^{2} + (- 3 b)^{2} + (- 2 c)^{2} -$

$2 a \times (- 3 b) - (- 3 b) \times (- 2 c) - (- 2 c) \times 2 a)$

Using Identity,

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

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

Comparing given equation with identity we get,

$(2 a)^{3} + (- 3 b)^{3} + (- 2 c)^{3} - 3 \times 2 a \times (- 3 b) \times (- 2 c)$

$= 8 a^{3} - 27 b^{3} - 8 c^{3} - 36 a b c$

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