We start with the given equation:

$(a + b + c)(ab + bc + ca) = abc$

Step 1: Expand $(a + b + c)^3$

The expression $(a + b + c)^3$ can be expanded as:

$(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + b^2a + b^2c + c^2b + c^2a + a^2c) + 6abc$

Alternatively, it can be written using the identity:

$(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a + b + c)(ab + bc + ca) - 3abc$

Step 2: Use the given equation

From the problem, we know:

$(a + b + c)(ab + bc + ca) = abc$

Substituting this into the expansion:

$(a + b + c)^3 = a^3 + b^3 + c^3 + 3 \cdot abc - 3 \cdot abc$

This simplifies to:

$(a + b + c)^3 = a^3 + b^3 + c^3$

Thus, the result for $(a + b + c)^3$ depends only on the individual cubes $a^3$, $b^3$, and $c^3$, with no additional terms.

Would you like a further breakdown or examples of how this might apply? Here are five questions to consider:

1. What specific values for $a, b, c$ satisfy the given condition?
2. Can we derive a similar equation if $(a + b + c)$ and $(ab + bc + ca)$ follow different relationships?
3. What happens if $a, b, c$ are negative or complex numbers?
4. How does this result relate to symmetric polynomial identities?
5. What are the implications for roots of polynomials with $a, b, c$ as roots?

Tip: Symmetric identities often simplify polynomial expressions; try to group terms by symmetry for clarity.