The question only asks to find the values of $a$ for which the inverse of $f(x)=x^3+ax^2+3x+2$ exists. To solve for the set of values of $a$, we can just set $f'(x)\ge0$ such that the equality only holds at discrete points. Then solving inequality, using the discriminant gives that $|a|\lt3$.
I read about Series Reversion, and tried to apply it for this polynomial but it is not clear as to how to coefficents of the inverse power series solution calculated here. It would be great if someone could illustrate it with this example at hand, and explain the computation of the coefficients of the inverse power series solution in the link (boldfaced). Thanks
Update $1$:
Let's simplify the problem, say find the inverse of $f(x)=x^3+3x+2$. So how to go about, can we comment on the degree of $f^{-1}(x)$ if its a polynomial function which it is unlikely to be. Besides, since we know its roots as it factorizes as $(x+1)(x^2+2)$, how can that be helpful in determining the inverse.