<s>[INST] Prove that if $a$ and $b$ are nonnegative real numbers, then $(a^7+b^7)(a^2+b^2) \ge (a^5+b^5)(a^4+b^4)$ Prove that if $a$ and $b$ are nonnegative real numbers, then $(a^7+b^7)(a^2+b^2) \ge (a^5+b^5)(a^4+b^4)$
My try
My book gives as a hint   to move everything to the left hand side of the inequality and then factor and see what I get in the long factorization process and to lookout  for squares. 
So that's what I have tried:
\begin{array}
((a^7+b^7)(a^2+b^2) &\ge (a^5+b^5)(a^4+b^4) \\\\
(a^7+b^7)(a^2+b^2)-(a^5+b^5)(a^4+b^4) &\ge 0 \\\\
(a+b)(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)(a^2+b^2)-(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4)(a^4+b^4) &\ge 0 \\\\
(a+b)\left[(a^6-a^5b+a^4b^2-a^3b^3+a^2b^4-ab^5+b^6)((a+b)^2-2ab)-(a^4+b^4)(a^4-a^3b+a^2b^2-ab^3+b^4)\right] &\ge 0 
\end{array}
Now it's not clear what I have to do next.I am stuck.
Note: My book doesn't teach any advanced technique for solving inequality as AM-GM ,Cauchy inequality etc.. 
 [/INST] \begin{align*}
(a^7 + b^7)(a^2 + b^2) - (a^5 + b^5)(a^4 + b^4) &= a^7 b^2 + a^2 b^7 - a^5 b^4 - a^4 b^5 \\
&= a^2 b^2 \big(a^5 + b^5 - a^3 b^2 - a^2 b^3\big) \\
&= a^2 b^2 \big(a^2 (a^3 - b^3) + b^2 (b^3 - a^3)\big) \\
&= a^2 b^2 (a^2 - b^2)(a^3 - b^3)
\end{align*}
Now regardless of how $a$ and $b$ are related to each other, the two difference terms above have the same sign.
</s>