`(x+3)/(x^2-2x-8)-(x-5)/(x^2-12x+32)` Perform the indicated operation(s) and simplify

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To be able to perform the indicated operation(s) on `(x+3)/(x^2-2x-8)-(x-5)/(x^2-12x+32)` , we have to express them as similar fractions.

Apply factoring on each expression on the denominator side.




 `x^2-12x+32=(x-4)(x-8) `

Determine the LCD by getting the product of the distinct factors from denominator side of each term.

Thus, `LCD =(x+2)(x-4)(x-8)`


                  `= x^3-2x^2-8x-8x^2+16x+64 `


Express each term by the LCD. Multiply top and bottom of each term by the missing factor.

First term:

`(x+3)/(x^2-2x-8) =(x+3)/((x+2)(x-4)) `

               ` =(x+3)/((x+2)(x-4))*(x-8)/(x-8)`

               ` =((x-8)(x+3))/((x+2)(x-4)(x-8)) `   

               ` =(x^2-5x-24)/(x^3-10x^2+8x+64)`

Second term:

`(x-5)/(x^2-12x+32) =(x-5)/((x-4)(x-8))`

                  `=(x-5)/((x-4)(x-8)) *(x+2)/(x+2) `


                  ` =(x^2-5x+2x-10)/(x^3-10x^2+8x+64)`


Applying the equivalent fraction in terms of LCD, we get:


`=(x^2-5x-24)/(x^3-10x^2+8x+64) -(x^2-3x-10)/(x^3-10x^2+8x+64)`

`=((x^2-5x-24) -(x^2-3x-10))/(x^3-10x^2+8x+64)`

`=(x^2-5x-24 -x^2+3x+10)/(x^3-10x^2+8x+64)`

`=(x^2-x^2-5x+3x-24+10)/(x^3-10x^2+8x+64) `


`=(-2x-14)/(x^3-10x^2+8x+64) or -(2x+14)/(x^3-10x^2+8x+64) `

Final answer:


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