You can rewrite your function as
[tex]f(x) = x^3(ax^3+bx+c)[/tex]
This implies that
[tex]f(x)=0 \iff x^3=0\quad\lor\quad ax^3+bx+c=0[/tex]
Now, we have [tex]x^3=0\iff x=0[/tex], so it counts as a solution.
On the other hand, depending on the coefficient a, b and c, the cubic equation
[tex]ax^2+bx+c=0[/tex]
can have either one or three solutions.
So, we have the solution x=0, and then one or three solutions coming from the cubic part. The equation as a whole thus have either two or four solutions, depending on the coefficients.
