Are you starting wiht your second expression (the one made up of 4 rational expressions added together) and getting them over a common denominator as I suggested earlier? Or starting with your first expression and trying to turn it into your second? The first way I suggested is so much easier.
I haven't heard it called that before....but anyway you'll need a term like (Ax+B)/(x^2+1) and also (Cx+D)/(x^2+1)^2.
Does that make sense to you?
I still think it is sooooo much easier to start with the second one and add terms by getting a common denominator. It would be different if you weren't given the second form and had to do it your way.