(5x²+17x+15)/((x+2)²(x+1))=A/(x+2)+B/(x+2)²+C/(x+1).
5x²+17x+15=A(x+1)(x+2)+B(x+1)+C(x+2)².
5x²+17x+15=A(x²+3x+2)+B(x+1)+C(x²+4x+4).
Equating coefficients:
x²: A+C=5, C=5-A
x: 3A+B+4C=17=3A+B+20-4A, A-B=3, B=A-3
constant: 2A+B+4C=15=2A+A-3+20-4A=-A+17, A=2, B=-1, C=3.
Partial fractions are 2/(x+2)-1/(x+2)²+3/(x+1).