You can write it like those too to help you:

You know that there are two vertical asymptotes at x = 5 and x = -3, then the graph also tends to the line y = x as x becomes large.

Since there are two asymptotes, the graph is in three parts and as similar graphs, the central part is fairly similar to the numerator, that is a cubic.

Now what you can do is try out what is the value of y slightly to the left and slightly to the right of the asymptotes to know the shape of the graph, whether it's ascending or descending.