The problem asks to simplify the expression $bc(b - c) + ca(c - a) + ab(a - b)$.

1. Problem Description

The problem asks to simplify the expression $bc(b - c) + ca(c - a) + ab(a - b)$.

2. Solution Steps

First, expand the expression:

$bc(b - c) + ca(c - a) + ab(a - b) = b^2c - bc^2 + c^2a - ca^2 + a^2b - ab^2$

Rearrange the terms:

$b^2c - bc^2 + c^2a - ca^2 + a^2b - ab^2 = b^2c - ab^2 + a^2b - ca^2 + c^2a - bc^2$

Factorize by grouping:

$b^2c - ab^2 + a^2b - ca^2 + c^2a - bc^2 = b^2(c-a) + b(a^2 - c^2) + ac(c-a) + (a - c)(-ac) = b^2(c-a) - b(c^2 - a^2) + ac(c-a)$

$b^2(c-a) - b(c - a)(c + a) + ac(c - a) = (c-a)[b^2 - b(c+a) + ac]$

$(c-a)[b^2 - bc - ba + ac] = (c-a)[b(b-c) - a(b-c)] = (c-a)(b-a)(b-c)$

Rearranging the terms:

$(c-a)(b-a)(b-c) = -(a-c)(-(a-b))(b-c) = (a-c)(a-b)(b-c)$

$(a-b)(b-c)(a-c) = (a-b)(b-c)(c-a)(-1) = -(a-b)(b-c)(c-a)$

Let's rearrange the terms to get $-(a-b)(b-c)(c-a) = (b-a)(c-b)(a-c)$

$b^2c - bc^2 + c^2a - ca^2 + a^2b - ab^2 = -(a-b)(b-c)(c-a)$

$= - (a-b)(bc - c^2 - b a + a c) = - (abc - ac^2 - a^2b + a^2c - b^2c + bc^2 + ab^2 - abc)$

$= -(- ac^2 - a^2b + a^2c - b^2c + bc^2 + ab^2)$

$= ac^2 + a^2b - a^2c + b^2c - bc^2 - ab^2$

Rearranging the original expression:

$= -(a-b)(b-c)(c-a)$

3. Final Answer

$-(a-b)(b-c)(c-a)$