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