We need to finish the decomposition

Two coefficients can be easily obtained using the cover-up method.

Now we need to obtain four equations to determine the remaining constants. The standard procedure is to use the multiplication method based on the general decomposition, where we multiply by the denominator. We use the knowledge of two of the constants.

Now we are supoosed to multiply out terms on the right and rewrite the right hand-side as a polynomial. By comparing coefficients one gets those equations. You are welcome to try, we will explore simpler alernatives here. First, one equation can be obtained using the limit trick, where we multiply the original equation by y and then pass to infinity.

One down, three to go, we try the plug-in trick and substitute some nice values for y into (*). We know that there is no point in substituting root, since that is equivalent to the cover-up trick. So we try some other nice numbers, then solve the resulting equations.

Since the equations had a certain symmetry to them, we used elimination, you are welcome to solve those four equations in any way you prefer. We are done