`(x^3-x+3)/(x^2+x-2)`
Divide by applying long division method,
`(x^3-x+3)/(x^2+x-2)=(x-1)+(2x+1)/(x^2+x-2)`
Now continue with the partial fraction of the remainder expression,
`(2x+1)/(x^2+x-2)=(2x+1)/(x^2-x+2x-2)`
`=(2x+1)/(x(x-1)+2(x-1))`
`=(2x+1)/((x-1)(x+2))`
Let `(2x+1)/(x^2+x-2)=A/(x-1)+B/(x+2)`
`(2x+1)/(x^2+x-2)=(A(x+2)+B(x-1))/((x-1)(x+2))`
`(2x+1)/(x^2+x-2)=(Ax+2A+Bx-B)/((x-1)(x+2))`
`:.(2x+1)=Ax+2A+Bx-B`
`2x+1=x(A+B)+2A-B`
Equating the coefficients of the like terms,
`A+B=2` ---- equation 1
`2A-B=1` --- equation 2
Add the equation 1 and 2,
`A+2A=2+1`
`3A=3`
`A=1`
Plug the value of A in the equation 1,
`1+B=2`
`B=2-1`
`B=1`
`:.(x^3-x+3)/(x^2+x-2)=x-1+1/(x-1)+1/(x+2)`
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