Find the value

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

Using the identity $(p+q)^{2}=p^{2}+q^{2}+2 p q$

We get,

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

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

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

Taking (a + b) common

$=(a+b)(a+b+2 c)$

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