# Multiple Partial Fractions

Algebra Level 4

$\large \frac{x^3+x^2+x+k}{(x+k)^2 (x^2+k^2)}$

Given that $$k\in\mathbb{Z}^+$$ is a constant and the expression above decomposes into partial fractions of

$\frac{C_{k,1}}{x+k}+\frac{C_{k,2}}{(x+k)^2}+\frac{C_{k,3}x+C_{k,4}}{x^2+k^2}$

where $$C_{k,1}$$, $$C_{k,2}$$, $$C_{k,3}$$, and $$C_{k,4}$$ are constants. Determine the value of $$\displaystyle \sum_{k=2}^{50} {C_{k,2}}-25$$.

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