First we need to find the linear factors of the denominator (note that 2 of these factors only exist as complex values):
#x^3+x=x(x^2+1)=color(blue)(x(x+i)(x-i))#
We need to find values #A, B, and C# such that
#(5x^2+7x+3)/(x^3+x)=A/x +B/(x+i)+C/(x-i)#
#color(white)("XXXXX")=(A(x+i)(x-i)+B(x)(x-i)+C(x)(x+i))/(x(x+i)(x-i))#
#color(white)("XXXXX")=(Ax^2+A+Bx^2-Bix+Cx^2-Cix)/(x^3+x)#
#rarr5x^2+7x+3=((A+B+C)x^2+(-Bi+Ci)x+A#
#rarrcolor(white)("XXXXX"){([1]color(white)("X")A+B+C=5),([2]color(white)("X")-Bi+Ci=7),([3]color(white)("X")A=3):}#
Combining [1] and [3] we have
#color(white)("XXXXX")[4]color(white)("X")B+C=2#
and from [2] we can
#color(white)("XXXXX")[5]color(white)("X")B-C=7i#
Adding [4] and [5] then dividing by 2:
#color(white)("XXXXX")[6]color(white)("X")B=1+7/2i#
Then substituting back into [4]
#color(white)("XXXXX")C=1-7/2i#
