Determine the domain of the function f(x)=sqrt of x+3/(x+8)(x-2)

Unclear.  Did you mean

f(x)= [sqrt of x+3]
       
          (x+8)

or did you mean
                               3
f(x)= [sqrt of x] +
                              x+8

I will focus on the first possibility.  The domain of sqrt(x+3) is [-3,infinity).
The domain of    x+3
                          is   "x is not equal to -8."
                            x+8

Then the overall domain would be simply [-3, infinity).