Question: How to write $f(x) = x^3 + 2x$ in 'completing the cubic' form [like we do in 'completing the square' for quadratic function]

Answer: 
You could write it as:

$x^3+2x=x^3+3x^2+3x+1-3x^2-x-1=(x+1)^3-3x^2-x-1$

I doubt, that there is a specific method to complete a cubic, in the sense, that you get it in the form $x^3+2x=a(x+e)^3+f$

Edit: In fact, if you write the equation above as

$x^3+2x=ax^3+3aex^2+3e^2x+e^3+f$

and now compare the coefficients on both sides, we see $a=1$. Also $e=0$, since there is no $x^2$ on the RHS. 
This leads to $x^3+f$ on the LHS, which is already wrong, so no such form exists here.
If the idea is isolate $x$ Note that
$$y=x^3+2x\implies x^3+2x-y=0$$
In the last equatión use the cubic formula or Cardano fórmula
[Cubic (Cardano) Formula](https://en.m.wikipedia.org/wiki/Cubic_function)