Factorize
$9 a^{2} + 4 b^{2} + 16 c^{2} + 12 a b - 16 b c - 24 c a$

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Solution

$9 a^{2} + 4 b^{2} + 16 c^{2} + 2 a b - 16 b c - 24 c a$
$=$ $(3 a)^{2} + (2 b)^{2} + (- 4 c)^{2} + 2 (3 a) (2 b) + 2 (2 b) (- 4 c) + 2 (- 4 c) (2 a)$
Suitable identities is $(x + y + z)^{3} = x^{3} + y^{3} + z^{3} + 2 x y + 2 y z + 2 x z$
Therefore, $(3 a)^{2} + (2 b)^{2} + (- 4 c)^{2} + 2 (3 a) (2 b) + 2 (2 b) (- 4 c) + 2 (- 4 c) (2 a)$
$=$ $(3 a + 2 b - 4 c)^{2}$

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