<s>[INST] Proving that: $ (a^3+b^3)^2\le (a^2+b^2)(a^4+b^4)$ This problem is from Challenge and Thrill of Pre-College Mathematics:
Prove that $$ (a^3+b^3)^2\le (a^2+b^2)(a^4+b^4)$$
It would be really great if somebody could come up with a solution to this problem.
 [/INST] \begin{eqnarray*}
\color{blue}{a^2b^2(a-b)^2} \geq 0 \\
\color{red}{a^6} +\color{blue}{a^4b^2+a^2 b^4} +\color{red}{b^6} \geq \color{red}{a^6} +\color{blue}{2 a^3b^3} +\color{red}{b^6} \\
(a^2+b^2)(a^4+b^4) \geq (a^3+b^3)^2.
\end{eqnarray*}
</s>