(5x+9)/(x(x+3)(x-4))=A/x+B/(x+3)+C/(x-4).
So, 5x+9=A(x²-x-12)+Bx(x-4)+Cx(x+3).
Matching coefficients:
Constant: -12A=9, A=-3/4
x: -A-4B+3C=5
x²: A+B+C=0, B+C=3/4, C=3/4-B.
3/4-4B+9/4-3B=5, -7B=2, B=-2/7, C=3/4+2/7=29/28.
Partial fractions: 29/(28(x-4))-2/(7(x+3))-3/(4x).