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 $a{b^2} + {a^2}b + {a^2}c + a{c^2} + {b^2}c + b{c^2} + 2abc$?

The polynomial in invariant under any permutation of $a,b,c$.
One way to attack this sort of polynomial is express it in terms
of its elementary symmetric polynomials and try to locate/detect any patterns that are useful.
We have $3$ variables, so we have $3$ elementary symmetric polynomials.
Let us call them $A,B,C$. They can be defined using following relations:
$$
(x-a)(x-b)(x-c) = x^3 - Ax^2 + Bx - C
\quad\iff\quad
\begin{cases}
A = a+b+c,\\
B = ab+bc+ca,\\
C = abc
\end{cases}$$
With help of $A,B,C$, we have:
$$\begin{align}
& {a^2}b + {a^2}c + a{c^2} + {b^2}c + b{c^2} + 2abc\\
= & ab(a+b) + bc(b+c)+ ca(c+a) + 2abc\\
= & ab(A-c) + bc(A-a)+ ca(A-b) + 2abc\\
= & (ab+bc+ca)A - abc\\
= & AB-C\\
= & A^3 - A\,A^2 + B\,A - C\\
= & (A-a)(A-b)(A-c)\\
= & (b+c)(c+a)(a+b)
\end{align}
$$
