With all these method, what should you do?

The first two methods are the same and very easy for determining the coefficients of the leading singularities at each zero of Q(x). Use them for this.

For the other coefficients, use whichever method seems easiest under the circumstances. The first is best if you can differentiate and evaluate reliably; the second if you can ide polynomials and evaluate carefully. The third is perhaps easiest. The fourth is purely algebraic and is the one I learned in high school. It has the most opportunities for error. It is tolerable if you love to solve linear equations. 

You should always check your answer by using the thirdmethod at x = 0 or 1 or infinity or some other new point after all coefficients are determined. Both sides of the equation must agree at the new point.