<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] $$\begin{array}{rrcl}
& (a^3+b^3)^2 &\le& (a^2+b^2)(a^4+b^4) \\
\iff& a^6 + 2a^3b^3 + b^6 &\le& a^6+a^2b^4+b^2a^4+b^6 \\
\iff& 2a^3b^3 &\le& a^2b^4+b^2a^4 \\
\iff& 2ab &\le& b^2+a^2 \\
\iff& 0 &\le& b^2-2ab+a^2 \\
\iff& 0 &\le& (b-a)^2 \\
\end{array}$$
</s>