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

Since the left side does not depend on changing signs of our variables, it's enough to prove our inequality for non-negatives $a$ and $b$.
Now, by C-S $$4(a^6+b^6)=2\cdot(1^2+1^2)(a^6+b^6)\geq2(a^3+b^3)^2.$$
Thus, it's enough to prove that:
$$2(a^3+b^3)\geq(a+b)(a^2+b^2)$$ or
$$2(a^2-ab+b^2)\geq a^2+b^2$$ or
$$(a-b)^2\geq0$$ and we are done!
