Explanation:
I think you may have forgotten you cannot have a sqrt of a negative number ....
The domain is the set of x values the function can have
all x EXCEPT values that cause
the denominator to = 0 OR be a sqrt 0f a negative number OR cause numerator to be sqrt of a negative
So for the numerator portion: x > 6 not allowed....
For the denominator:
sqrt ( x^2-4(x+3)^2 ) cannot equal 0
and 4 (x+3)^2 cannot = x^2
4x^2 +24x + 36 cannot = x^2
3x^2 + 24x + 36 cannot = 0
3 (x^2 + 8x+12) cannot = 0
x^2 + 8x + 12 cannot = 0
(x+2)(x+6) cannot = 0
x cannot = -2 or -6
then x^2-4(x+3)^2 <0 not allowed ( it would be sqrt of a negative)
x^2 - 4x^2 -24x-36 <0 not allowed
-3x^2 -24x -36 <0 not allowed
-3 ( x^2 +8x+12) <0
(x+6)(x+2) < 0 not allowed <=====this is a dome shaped parabola (see attached image)
so greater than -2 or less than -6 not allowed
Domain is then -2>x>-6 ( or re-written as -6<x<-2 )
Is what I find -6 and -2 cannot be included....the denom would be 0 (as found above) I do not know where -3 comes from ....
I submitted this to Wolframalpha and this is the answer it found too...so I think your textbook is in error.....