First you need to find limit of `f(x)` when `x->-6.`

`lim_(x->-6)(4x^3+24x^2+9x+54)/(x+6)=`

Now we divide `4x^3+24x^2+9x+54` by `x-6` so we get

`lim_(x->-6)(4x^2+9)=4(-6)^2+9=153`

So the limit of this function at -6 is 153 so in order to make the function continuous at -6 the other part of the function must also...

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First you need to find limit of `f(x)` when `x->-6.`

`lim_(x->-6)(4x^3+24x^2+9x+54)/(x+6)=`

Now we divide `4x^3+24x^2+9x+54` by `x-6` so we get

`lim_(x->-6)(4x^2+9)=4(-6)^2+9=153`

So the limit of this function at -6 is 153 so in order to make the function continuous at -6 the other part of the function must also have value of 153 at x=-6. That is when we put x=-6 in `4x^2+x+A` we must get 153.

`4(-6)^2-6+A=153`

`144-6+A=153`

` `

`A=15` **<-- A must be equal to 15.**

Your function is

`f(x)={((4x^3+24x^2+9x+54)/(x+6) if x<-6),(4x^2+x+15 if x>=-6):}`

Graph:

Blue and red curve meet at x=-6.