2
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For example, $\dfrac{1}{(x^2+1)^2}$

This may be a dumb question as I'm pretty sure you can't, but then why not?

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    $\begingroup$ It IS a partial fraction. $\endgroup$ – Przemysław Scherwentke Apr 17 '18 at 20:39
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    $\begingroup$ @PrzemysławScherwentke I think you should expand that into an answer, quoting the definition of partial fraction decomposition that makes it clear that over the reals irreducible quadratic denominators are allowed. $\endgroup$ – Ethan Bolker Apr 17 '18 at 20:42
  • $\begingroup$ @EthanBolker I would rather wait for clarification of the question. We know that both forms are partial fractions, but we don't know OP's expectations. $\endgroup$ – Przemysław Scherwentke Apr 17 '18 at 20:46
  • $\begingroup$ Are the complex numbers allowed? $\endgroup$ – user Apr 17 '18 at 20:58
  • $\begingroup$ If you want to confine yourself to real coefficients, then it depends on whether or not B and C have the same sign. $\endgroup$ – herb steinberg Apr 17 '18 at 21:50

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