Precalculus, Chapter 7, 7.4, Section 7.4, Problem 39

(x^2+5)/((x+1)(x^2-2x+3))
Let(x^2+5)/((x+1)(x^2-2x+3))=A/(x+1)+(Bx+C)/(x^2-2x+3)
(x^2+5)/((x+1)(x^2-2x+3))=(A(x^2-2x+3)+(Bx+C)(x+1))/((x+1)(x^2-2x+3))
(x^2+5)/((x+1)(x^2-2x+3))=(Ax^2-2Ax+3A+Bx^2+Bx+Cx+C)/((x+1)(x^2-2x+3))
:.(x^2+5)=Ax^2-2Ax+3A+Bx^2+Bx+Cx+C
x^2+5=(A+B)x^2+(-2A+B+C)x+3A+C
equating the coefficients of the like terms,
A+B=1
-2A+B+C=0
3A+C=5
Now let's solve the above three equations to find the values of A,B and C,
Express C in terms of A from the third equation,
C=5-3A
Substitute the above expression of C in second equation,
-2A+B+5-3A=0
-5A+B+5=0
-5A+B=-5
Now subtract the first equation from the above equation,
(-5A+B)-(A+B)=-5-1
-6A=-6
A=1
Plug the value of A in the first and third equation to get the values of B and C,
1+B=1
B=1-1
B=0
3(1)+C=5
C=5-3
C=2
:.(x^2+5)/((x+1)(x^2-2x+3))=1/(x+1)+2/(x^2-2x+3)
Now let's check it algebraically,
1/(x+1)+2/(x^2-2x+3)=(1(x^2-2x+3)+2(x+1))/((x+1)(x^2-2x+3))
=(x^2-2x+3+2x+2)/((x+1)(x^2-2x+3))
=(x^2+5)/((x+1)(x^2-2x+3))
Hence it is verified.