What you were doing there is some kind of curve sketiching but without the powerful tools of calculus itself. But the things you denoted there are quite useful aswell.
First of all you know something about the asymptotic behavior of the function in two ways. So you can just say that the function will go to infinity while the x-value is decreasing and it will go to negative infinity while the x-value is increasing. These you can just derivate from the oblique asymptote you figured out. So now you know how the function will develop with the x argument.
As a next step concentrate on the oblique asymptote. You know the y-value will reach infinity in the near of this asymptote. When you put this together with the roots of the polynomial you can go further with the form of the graph.
Both roots are smaller than the x-value of the oblique asymptote. Also the function goes for infinity for big negative x-values. So the graph will fall while you going to the first root but right after this it have to rise against since the graph has to reach the next root AND has to go for infinity at the oblique asymptote.
On the other hand the right side of the graph never hits the x-axis hence it goes from negative infinity up while the x-value is decreasing. So if the function would now go for infinity again there would be another interception with the x-axis. So it goes to negative infinity.
To put it all together: From the asymptotes you know how the function behaves at critical points such as infinity, negative infinity and at poles. Now you just focus on the values of the roots and compare them with the poles. From there on you can sketch the graph in a way of thinking how it develops at several x-values.