<s>[INST] Proving the inequality $4(a^6+b^6) \ge (a+b)(a^2+b^2)(a^3+b^3)$ Prove that $4(a^6+b^6) \ge (a+b)(a^2+b^2)(a^3+b^3)$. When does the inequality hold?
I really don't know how to prove the inequality and would like to know how.
I mainly tried to factorise the LHS-RHS fully but I could never properly do it: 
https://imgur.com/user/Khansis/favorites/folder/7408635/math
 [/INST] It is $$4(a^6+b^6)-(a+b)(a^2+b^2)(a^3-b^3)=(a-b)^2(a^2+b^2)(3a^2+5ab+3b^2)\geq 0$$
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