My attempt at a solution :
Okay, so I can get the solution by working backwards ( working out each side separately, cubing it, finding the final expression, and then I can just work my way back up to show that the original inequality was true) , but this is somewhat messy, and the question clearly asks me to use the factorization of $x^3 -y^3$, which I see no point to apply in my solution.
Could someone point me towards how to add this in my answer?
Hint:
Let $x^3 = a^3$ and $y^3= a^3-h$. Note that $x^3-y^3=h$. Factorize $x^3-y^3$ and use the given constraints on $h$ to prove the inequalities. $$h=x^3-y^3=(x-y)(x^2+xy+y^2)=(a-\sqrt[3]{a^3-h})(a^2+a(a^3-h)^{1/3}+(a^3-h)^{2/3}).$$