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

Since $a + b ≥ 0$, we can apply A.M.-G.M. inequality, I tried to apply the inequality, but wasn't able to reach a conclusive decision. How can I solve it using A.M.-G.M. inequality, or by any other way which is much easier than prior method.
 [/INST] We need to prove that:
$$(a^2+b^2)^3+32(a+b)^2(a^2-ab+b^2)\geq32(a+b)(a^2-ab+b^2)ab$$ and since $a+b\geq0,$ it's enough to prove our inequality for $ab\geq0,$ which gives that $a$ and $b$ are non-negatives.
Now, by AM-GM 
$$(a^2+b^2)^3+32(a+b)^2(a^2-ab+b^2)\geq2\sqrt{(a^2+b^2)^3\cdot32(a+b)^2(a^2-ab+b^2)}.$$
Thus, it's enough to prove that:
$$2\sqrt{(a^2+b^2)^3\cdot32(a+b)^2(a^2-ab+b^2)}\geq32(a+b)(a^2-ab+b^2)ab$$ or
$$(a^2+b^2)^3\geq8(a^2-ab+b^2)a^2b^2.$$
Let $a^2+b^2=2kab$. 
Thus, by AM-GM again $k\geq1$ and we need to prove that 
$$(2k)^3\geq8(2k-1)$$ or 
$$k^3-2k+1\geq0$$ or
$$k^3-k^2+k^2-k-k+1\geq0$$ or $$(k-1)(k^2+k-1)\geq0,$$ which is true for $k\geq1.$
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