Precalculus, Chapter 7, 7.4, Section 7.4, Problem 51

(5-x)/(2x^2+x-1)
=(5-x)/(2x^2+2x-x-1)
=(5-x)/((2x(x+1)-1(x+1)))
=(5-x)/((2x-1)(x+1))
Let (5-x)/(2x^2+x-1)=A/(2x-1)+B/(x+1)
(5-x)/(2x^2+x-1)=(A(x+1)+B(2x-1))/((2x-1)(x+1))
(5-x)/(2x^2+x-1)=(Ax+A+2Bx-B)/((2x-1)(x+1))
(5-x)/(2x^2+x-1)=(x(A+2B)+A-B)/((2x-1)(x+1))
:.(5-x)=x(A+2B)+A-B
A+2B=-1
A-B=5
solving the above two linear equations to get A and B ,
subtracting the second equation from the first equation,
3B=-1-5
3B=-6
B=-6/3
B=-2
Plug the value of A in second equation,
A-(-2)=5
A+2=5
A=5-2
A=3
:.(5-x)/(2x^2+x-1)=3/(2x-1)-2/(x+1)