Find the function $g(a)$ which is the difference between the maximum & min values for $f(x)=(a-\frac{1}{a}-x)(4-3x^2)$ for $a \in R$

Question

I want to know if there are "shorter" ways to approach this question as my solution is way lengthier, and I wasn't able to complete it either, and I feel like i was not able to manipulate the algebra well.

1. I have substituted $a-\frac{1}{a}=k$, but this didn't make the solution less lengthier.
2. I have found the critical points as $x= \frac{k \pm \sqrt{k^2 + 4}}{3}$.
3. I have substituted x in the original function but was unable to find a nice expression, and hence wasn't able to compute the difference of the maxima and minima.

The answer is given as $ g(a) = \frac{4}{9} \cdot (a + \frac{1}{a})^3$ which looks very simple, so I was wondering if I could approach the problem in a nicer way to get to it.

Side note: I have added the algebra-precalculus tag because I am facing difficulties in the algebraic manipulation part.