x^4+5x³+6x²=x²(x²+5x+6)=x²((x²+2x)+(3x+6))=x²(x(x+2)+3(x+2))=
=x²(x+2)(x+3)
(x+1)/(x^4+5x³+6x²)=A/x²+B/(x+2)+C/(x+3)+D/x
A*(x²+5x+6)+B*(x³+3x²)+C*(x³+2x²)+D*(x³+5x²+6x)=x+1
x³*(B+C+D)+x²*(A+3B+2C+5D)+x*(5A+6D)+6*A=x+1
{B+C+D=0
{A+3B+2C+5D=0
{5A+6D=1
{6A=1⇒A=1/6
подставим в 3
5/6+6D=1
6D=1-5/6=1/6
D=1/6:6=1/36
подставим в 1 и 2
{B+C=-1/36/*(-2)⇒-2B-2C=2/36
{3B+2C=-11/36
прибавим
B=-9/36=-1/4
C=-1/36+1/12=-1/36+3/36=2/36=1/18
Получили
1/(6х²)-1/(4(х+2))+2/(9(x+3))+1/(36х)=(х+1)/(х^4+5x³+6x²)