Question
Factorize: $a^2+b^2+2(a b+b c+c a)$
✓
Answer
$=a^2+b^2+2 a b+2 b c+2 c a$
Using the identity $(p+q)^2=p^2+q^2+2 p q$
We get,
$=(a+b)^2+2 b c+2 c a$
$=(a+b)^2+2 c(b+a)$
$\text { Or }(a+b)^2+2 c(a+b)$
Taking $(a+b)$ common
$=(a+b)(a+b+2 c)$
$\therefore a^2+b^2+2(a b+b c+c a)=(a+b)(a+b+2 c)$
Using the identity $(p+q)^2=p^2+q^2+2 p q$
We get,
$=(a+b)^2+2 b c+2 c a$
$=(a+b)^2+2 c(b+a)$
$\text { Or }(a+b)^2+2 c(a+b)$
Taking $(a+b)$ common
$=(a+b)(a+b+2 c)$
$\therefore a^2+b^2+2(a b+b c+c a)=(a+b)(a+b+2 c)$
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