Find how many solutions this ln polinomial equation has

[transgalactic]

$$2x^3lnx+x^2-4x+a=0$$
i thought of building a taylor series around 0 for ln
but ln(0) is undefined

??

 

[rootX]

I would find how many max min it has then knowing the each max min position I can determine the number of solutions.
Take care of the x= 0 too.