Let's factorise the denominator x^5+7x^3=x^3(x^2+7)x5+7x3=x3(x2+7)
So (x^4+6)/(x^3(x^2+7))=A/x+B/x^2+C/x^3+(Dx+E)/(x^2+7)x4+6x3(x2+7)=Ax+Bx2+Cx3+Dx+Ex2+7
=(Ax^2(x^2+7)+Bx(x^2+7)+C(x^2+7)+(Dx+E)(x^2+7))/(x^3(x^2+7))=Ax2(x2+7)+Bx(x2+7)+C(x2+7)+(Dx+E)(x2+7)x3(x2+7)
x^4+6=Ax^2(x^2+7)+Bx(x^2+7)+C(x^2+7)+(Dx+E)x^3x4+6=Ax2(x2+7)+Bx(x2+7)+C(x2+7)+(Dx+E)x3
x=0x=0=>⇒6=7C6=7C=>⇒C=6/7C=67
coefficients x^4x4=>⇒1=A+D1=A+D
coefficients x^3x3=>⇒0=B+E0=B+E
coefficients x^2x2=>⇒0=7A+C0=7A+C
coefficients of xx=>⇒0=7B0=7B
E=0E=0
A=-C/7=-6/49A=−C7=−649
B=0B=0
D=1-A=1+6/49=55/49D=1−A=1+649=5549
Finally we have
(x^4+6)/(x^3(x^2+7))=(-6/49)/x+(6/7)/x^3+(55/49x)/(x^2+7)x4+6x3(x2+7)=−649x+67x3+5549xx2+7
