#### Answer

$ f’(x) =-\frac{x^{2}+8x+3}{(x^{2}+x+1)^{2}} $

#### Work Step by Step

$Quotient$ $Rule:$
$ (\frac{f}{g})’ = \frac{gf’-fg’}{g^{2}} $
$ f(x) = \frac{x+4}{x^{2}+x+1} $
$ f’(x) = \frac{(x^{2} +x+1)(1)-(x+4)(2x+1)}{(x^{2}+x+1)^{2}} $
$ f’(x) =\frac{-x^{2}-8x-3}{(x^{2}+x+1)^{2}} $