Factoring an expression with three variables the expression is  $(a-b)c^3+(b-c)a^3-(a-c)b^3$
I've tried factoration by grouping, but it has been a trial and error situation. I don't know any strategies to factor expressions like that, so I'm basically just guessing.
There is a ton of questions like this one in the book I'm studying right now and I can't solve anything, it just feels like I have to guess since I don't know any method to do this.

One may first set the expression equal to $0$.
$$0=(a-b)c^3+(b-c)a^3+(c-a)b^3$$
One can then see that for this to hold, we have one solution $a=b$, $a=c$, and $b=c$.  Turning this into "factors" that we can use, we get, as a polynomial of $a$,
$$(a-b)(a-c)(b-c)P(a)=(a-b)c^3+(b-c)a^3+(c-a)b^3$$
whereupon one will find that
$$P(a)=a+b+c$$
Thus,
$$(a-b)c^3+(b-c)a^3+(c-a)b^3=(a-b)(a-c)(b-c)(a+b+c)$$
