How can we factorize $a{b^2} + {a^2}b + {a^2}c + a{c^2} + {b^2}c + b{c^2} + 2abc$? How can we factorize $a{b^2} + {a^2}b + {a^2}c + a{c^2} + {b^2}c + b{c^2} + 2abc$?

You just need to do a little rearrangement :
$$a{b^2} + {a^2}b + {a^2}c + a{c^2} + {b^2}c + b{c^2} + 2abc$$
         $$a^2b+a^2c+b^2c+bc^2+ab^2+abc+abc+ac^2$$
$$a^2(b+c)+bc(b+c)+ab(b+c)+ac(b+c)$$
$$(b+c)(a^2+bc+ab+ac)$$
$$(b+c)(a(a+b)+c(a+b))$$
$$(b+c)(a+b)(a+c)$$
$$(a+b)(b+c)(c+a)$$
 *And we are done !!!!!
