Can this Fourier transformed function be transformed into partial fraction?

Question

Hi I'm self learning stochastic process, I've come across a problem and found $$H=\frac{2+j(2w)}{(1-w^2)+j(2w)} \\ |H(iw)|^2=\frac{2^2+(2w)^2}{(1-w^2)^2+(2w)^2} \\$$ In attempt to find the power spectral density with partial fraction, $$S_y(w)=|H(iw)|^2 S_x(w)=\frac{2^2+(2w)^2}{(1-w^2)^2+(2w)^2} \frac{3}{2^2+w^2} \\ = \frac{12(1+w^2)}{(w^2+1)^2(w^2+4)} = \frac{A}{(w^2+1)^2}+\frac{B}{(w^2+4)} \\ A = \frac{12(1+w^2)}{w^2+4} = 0, w=\pm\sqrt{-1}= \pm j \\ B=\frac{12(1+w^2)}{(w^2+1)^2} = \frac{12(1-4)}{(-4+1)^2}=\frac{-36}{9}=-4, w^2+4=0, w=\pm2j$$ Is complex $w$ even making any sense? Because I thought w is part of $iw$, then it should be real if that makes sense.

Answer 1

$$\frac{12(1+w^2)}{(w^2+1)^2(w^2+4)}=\frac{12}{(w^2+1)(w^2+4)}=\frac{4}{w^2+1}-\frac{4}{w^2+4}$$