Consider the rational function $f(x)=\frac{x^2-3x-4}{x^2+1}$ my analysis prof gave us this definition of the partial fraction decomposition:
Given this definition I would argue that $f$ has no partial fraction decomposition since
$\deg(x^2-3x-4) = \deg(x^2+1)$
Is this correct or can you still find a representation and if so how?
EDIT:
By performing euclidian division I obtain:
$f(x)= 1 - \frac{3x+5}{x^2+1}$
Now I would have to find a decomposition of $\frac{-3x+5}{x^2+1}$ this clearly would be complex since $x^2+1=(x+i)(x-i)$ but the given exercise problem says that I should only consider decompositions in the real numbers. Does that mean there is none?