The denominator factors are x^2-k, so the poles are the eight values plus or minus the square roots of 1, 2, 3, and 4, not the four positive parameters themselves.  Multiplying by the common denominator gives the coefficient certificate in the submission.  At a pole square k, every cleared summand vanishes except the kth one; its residual is k times the product of k-j over j different from k.  The four residuals are nonzero, so none of the denominator zeros is a root of the cleared polynomial.  That polynomial has leading coefficient -2010 and degree-eight coefficient 4.  Vieta therefore gives the sum of all nine complex roots, with multiplicity, as 2/1005.
