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

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Solution

$(a + 2 b + 3 c)^{2} - (a - 2 b + 3 c)^{2} - 6 b^{2} - a b c$
$= a^{2} + 4 b^{2} + 9 c^{2} + 4 a b + 12 b c + 6 a c - (a^{2} + 4 b^{2} + 9 c^{2} - 4 a b - 12 b c + 6 a c) - 6 b^{2} - 9 a c$
$= a^{2} + 4 b^{2} + 9 c^{2} + 4 a b + 12 a c + 6 a c - a^{2} - 4 b^{2} - a c^{2} + 4 a b + 12 b c - 6 a c - 6 b^{2} - 9 b c$
$= 8 a b + 24 b c - 6 b^{2} - 9 b c$

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