Because the limit approaches when you substitute -4 and when you substitute you can apply Hospital's rule:
if
then
So:
The only reason you want to factor that is because both it and the denominator become 0 when x= -3. That means that x-(-3)= x+ 3 is a factor. Now just do a "long division" to find the other factor: x goes into times. Multiplying by x+ 3 we get
Now, x goes into -x times. Multiplying -x by x+ 3 we get .
Finally, x goes into -2x -2 times and, multiplying -2 by x+ 3 we get
.