Let real number  "t"  belongs to the range.  It means that

     = t    (1)

for some value of x.  Step by step, reduce equation (1) to standard form quadratic equation

     = ,

     = 0.    (2)


Equation (2) is a standard form quadratic equation   = 0  with coefficients

    a = 1,  b = -t,  c = 2-4t.


The condition that it has a real solution for   is this inequality for the discriminant


     >= 0,  or   >= 0,  or  t^2 + 16t -8 >= 0.


The roots of this quadratic equation for t are


     =  =  = -16.4853 (rounded)  and   =  =  = 0.4853 (rounded).


The discriminant is non-negative if and only if value of  "t"  is out of the roots' interval 


    t <=  = -16.4853  or  t >=  =0.48528.


Therefore, equation (1) has a real solution if and only if   < t <=    or   <= t < .


Thus the range of the function  f(x) = (x^2 + 2)/(x + 4) is the union of these intervals  (,]  and  [,).


ANSWER.  The range of the function    is the union of these intervals  (,]  and  [,).