I. I start with looking at xb-plane. Parameter a changes from -5 to 5.

Look.

We can see that the number of roots when b=0 is either one, two or three.

II. I continue by looking at the xc-plane. Again I change parameter a from -5 to 5.

Look.

We can see that if a=c=0 we have no roots. Else we have only one root.

III. Let us look now at xd-plane. Again, parameter a changes from -5 to 5.

How many roots do we have? Well, when d=0 we get ax^3+x^2+x=0. Thus, x(ax^2+x+1)=0. Therefore, for d=0 we get only one solution.

If a=0 as well then x^2+x=0. Thus, x(x+1)=0. Therefore, we get two solutions.

For negative a, we get three solutions.

It is interesting to mention that the graph always goes trough (0,0). Thus, (0,0) is solution independently of value of the parameters.

IV. In the end let us look at the xa-plane.

As line y=a changes values we get either zero or one solution.