I'm getting a slightly different answer: I don't come up with the , but I've got everything else. Any ideas what I could have missed?
I went the route and got and .
The steps the book follows confuse me, too. You can see them at calcchat.com|Textbook: Calculus, 8th Edition|Chapter 8|Section 5|Exercise 15.
Such as where that and came from in step 1.
Since the degree of the numerator is larger than the degree of the denominator you have to do a long division and divide the quadratic into the cubic before attempting to get a partial fraction decomposition. The result of the long division is
.
Now you can attempt to get a partial fraction decomposition ........ of .