Factorize the given polynomial expression

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

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Solution

Given That,
$a^{2} + b^{2} + 2 (a b + b c + c a)$

By Expanding we get,

$= a^{2} + b^{2} + 2 a b + 2 b c + 2 c a$

By Rearranging the terms we get,

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

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