Factorize $a^{2} + 4 b^{2} + 36 - 4 a b - 24 b + 12 a$

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Solution

$a^{2} + 4 b^{2} + 36 - 4 a b - 24 b + 12 a$ can be written as
$(a)^{2} + (- 2 b)^{2} + (6)^{2} + 2 (a) (- 2 b) + 2 (- 2 b) (6) + 2 (6) (a) = (a - 2 b + 6)^{2}$ or
$(- a)^{2} + (2 b)^{2} + (- 6)^{2} + 2 (- a) (2 b) + 2 (2 b) (- 6) + 2 (- 6) (- a) = (- a + 2 b - 6)^{2}$
That is $(a - 2 b + 6)^{2} = [(- 1) (- a + 2 b - 6)^{2}] = (- 1)^{2} (- a + 2 b - 6)^{2} = (- a + 2 b - 6)^{2}$

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