Example 6 Given $a+b+c=5$,
$$
\begin{array}{l}
a^{2}+b^{2}+c^{2}=15, a^{3}+b^{3}+c^{3}=47 . \\
\text { Find }\left(a^{2}+a b+b^{2}\right)\left(b^{2}+b c+c^{2}\right)\left(c^{2}+c a+a^{2}\right)
\end{array}
$$

the value.
【Analysis】Since the target algebraic expression is a symmetric algebraic expression, consider expressing it using $a+b+c$, $ab+bc+ca$, and $abc$. According to the problem, the values of $ab+bc+ca$ and $abc$ can be found using two common formulas.
Solution From formula (1), we know
$$
\begin{array}{l}
2(ab+bc+ca) \\
=(a+b+c)^{2}-\left(a^{2}+b^{2}+c^{2}\right)=10 \\
\Rightarrow ab+bc+ca=5 .
\end{array}
$$

From formula (2), we know
$$
\begin{array}{l}
47-3abc=5(15-5) \Rightarrow abc=-1 . \\
\text { And } a^{2}+ab+b^{2} \\
=(a+b)(a+b+c)-(ab+bc+ca) \\
=5(5-c)-5=5(4-c),
\end{array}
$$

Similarly, $b^{2}+bc+c^{2}=5(4-a)$,
$$
c^{2}+ca+a^{2}=5(4-b) \text {. }
$$

Therefore, the value of the required expression is
$$
\begin{array}{l} 
125(4-a)(4-b)(4-c) \\
= 125(64-16(a+b+c)+ \\
4(ab+bc+ca)-abc) \\
= 125(64-16 \times 5+4 \times 5-(-1)) \\
= 125 \times 5=625 .
\end{array}
$$