Transform the following sum into a product:

$$
(a-b)^{5}+(b-c)^{5}+(c-a)^{5}
$$
Let

$$
a-b=x, \quad b-c=y
$$

and from this

$$
c-a=-(x+y)
$$

so that the given sum takes the following form:

$$
x^{5}+y^{5}-(x+y)^{5}
$$

This latter expression, when the binomial fifth power is expanded, takes the following form.

$$
\begin{aligned}
& -5\left(x^{4} y+2 x^{3} y^{2}+2 x^{2} y^{3}+x y^{4}\right) \\
& =-5 x y\left(x^{2}+2 x^{2} y+2 x y^{2}+y^{3}\right) \\
& =-5 x y(x+y)\left(x^{2}+y^{2}+x y\right)
\end{aligned}
$$

If we now substitute $a-b$ and $b-c$ for $x$ and $y$ respectively, the sum transforms into the following product:

$$
5(a-b)(b-c)(c-a)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)
$$