For example, I am given that factorize:
$$a^2b+a^2c+ab^2+2abc+ac^2+b^2c+bc^2$$
So by the traditional method, we take the powers of $a$
$$=a^2(b+c)+a(b+c)^2+bc(b+c)$$
$$=(b+c)(a^2+ab+ac+bc)$$
Now the powers of $b$:
$$=(b+c)(b(a+c)+a(a+c))$$
$$=(b+c)(a+c)(a+b)$$
Which gives a neat factorization.
But my question is: Why does this method work at all? Any intuition? Any proof?