Proof: Let
$$f(a, b, c)=a^{4}+b^{4}+c^{4}+a b^{3}+b c^{3}+c a^{3}-2\left(a^{3} b+b^{3} c+c^{3} a\right)$$

Then
$$\begin{array}{l} f(a+t, b+t, c+t)=6\left(\sum a^{2}-\sum a b\right) t^{2}+3\left(\sum a^{3}+\sum_{\text {cr }} a^{2} b-2 \sum_{c y c} a b^{2}\right) t+ \\ a^{4}+b^{4}+c^{4}+a b^{3}+b c^{3}+c a^{3}-2\left(a^{3} b+b^{3} c+c^{3} a\right) \end{array}$$

Thus, it suffices to prove
$$\begin{array}{l} \Delta(a, b, c)=3\left(\sum a^{3}+\sum_{q c} a^{2} b-2 \sum_{q c} a b^{2}\right)^{2}-4 \cdot 6\left(\sum a^{2}-\sum a b\right) \cdot \\ \left(a^{4}+b^{4}+c^{4}+a b^{3}+b c^{3}+c a^{3}-2\left(a^{3} b+b^{3} c+c^{3} a\right)\right) \leqslant 0 \end{array}$$

Notice that $\Delta(a, b, c)=\Delta(a+t, b+t, c+t), t \in \mathbf{R}$, so it suffices to prove
$$\begin{array}{l} \Delta(a-c, b-c, 0) \leqslant 0 \Leftrightarrow \\ \left(3(a-c)^{2}(b-c)-6(a-c)(b-c)^{2}+3(a-c)^{3}+3(b-c)^{3}\right)^{2}- \\ 4\left(-3(a-c)(b-c)+3(a-c)^{2}+3(b-c)^{2}\right)\left((a-c)^{4}+(b-c)^{4}+\right. \\ \left.(a-c)^{3}(b-c)-2(a-c)(b-c)^{3}\right) \leqslant 0 \Leftrightarrow \\ -3\left((b-c)^{3}-3(a-c)^{2}(b-c)+(a-c)^{3}\right)^{2} \leqslant 0 \end{array}$$

The proposition is proved.