`(x^2-4x+7)/[(x+1)(x^2-2x+3)]=A/(x+1)+(Bx+C)/(x^2-2x+3)`

Multiply through by the LCD` (x+1)(x^2-2x+3)`

`x^2-4x+7=A(x^2-2x+3)+(Bx+C)(x+1)`

`x^2-4x+7=Ax^2-2Ax+3A+Bx^2+Cx+Bx+C`

`x^2-4x+7=(A+B)x^2+(-2A+C+B)x+(3A+C)`

Equate the coefficients of like terms. Then solve for A, B, and C.

`1=A+B`

`-4=-2A+B+C`

`7=3A+C`

`-4=-2A+B+C`

`-4=-2A+(1-A)+(7-3A)`

`-4=-6A+8`

`-12=-6A`

`2=A`

`1=A+B`

`1=2+B`

`-1=B`

`7=3A+C`

`7=3(2)+C`

`1=C`

`A=2, B=-1, C=1`

`(x^2-4x+7)/[(x+1)(x^2-2x+3)]=2/(x+1)-(x-1)/(x^2-2x+3)`

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