<s>[INST] If $a,b \in \mathbb{R}$ and $a+b\ge 0$, prove that $(a^2+b^2)^3\ge 32(a^3+b^3)(ab-a-b)$ 
If $a,b \in \mathbb{R}$ and $a+b\ge 0$, prove that $(a^2+b^2)^3\ge 32(a^3+b^3)(ab-a-b)$

This question in my opinion is difficult and have tried many things. I tried to expand both sides but that will not help since I will not be able to cancel anything out. I am also struggling with other methods like AM-GM since $a,b$ are not necessarily positive. Any help would be appreciated.
 [/INST] Assuming that $a,b > 0$, you want to prove that:
$$a^6+3a^4b^2+3a^2b^4+b^6+32a^4+32a^3b+32ab^3+32b^4 \geq 32 a^4b+ 32ab^4.$$
Now, show that $$a^6+a^4b^2+a^4b^2+a^4b^2+16a^4+16a^4+16a^3b+16a^3b \geq 32a^4b.$$
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