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

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

` `

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

` `

`((x+2)^2(3x-9-2x-4))/(x-3)^3=`

` ` `((x+2)^2(x-13))/(x-3)^3` **<-- This is your simplified expression.**

` `

As you can see there is no need to use quotient rule. Furthermore the exercise in the image doesn't say you should use quotient rule, it says that such expression arises in calculus when using quotient rule. To be more precise you would get such expression if you were to differentiate this function

`f(x)=(x+2)^3/(x-3)^2`

And you would need to use quotient rule in order to differentiate that function.

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