(x2-2x-9)/[x2+x+6)(x+1)]=(Ax+B)/(x2+x+6)+C/(x+1)
x2-2x-9=(Ax+B)(x+1)+C(x2+x+6)=Ax2+(A+B)x+B+Cx2+Cx+6C.
Equate like terms:
x2: A+C=1, A-1=1, A=2
x: A+B+C=-2, B+1=-2, B=-3;
constant: B+6C=-9, -3+6C=-9, 6C=-6, C=-1.
(x2-2x-9)/[x2+x+6)(x+1)]=(2x-3)/(x2+x+6)-1/(x+1).
CHECK
(2x-3)(x+1)-x2-x-6=2x2-x-3-x2-x-6=x2-2x-9