<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] You can simplify first:
$$4(\underbrace{a^6+b^6}_{(1)}) =4\require{cancel}\cancel{(a^2+b^2)}(a^4-a^2b^2+b^4)\ge (a+b)\cancel{(a^2+b^2)}(\underbrace{a^3+b^3}_{(2)}) \Rightarrow\\
4((a^2-b^2)^2+a^2b^2)\ge (a+b)^2((a-b)^2+ab) \Rightarrow \\
3(a^2-b^2)^2+ab(4ab-(a+b)^2)\ge 0 \Rightarrow \\
3(a-b)^2(a+b)^2-ab(a-b)^2\ge 0 \Rightarrow \\
(a-b)^2(\underbrace{3a^2+3b^2+5ab}_{(3)})\ge 0 \quad \color{green}\checkmark$$
The equality holds when $a=b$.
Note: 
1) For $(1),(2)$ it was used: $a^3+b^3=(a+b)(a^2-ab+b^2)$. 
2) For $(3)$, it was used AM-GM: $3(a^2+b^2)\ge 6|ab|\ge 5ab$.   
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