Answer
$h^{\prime}=-\displaystyle \frac{4x+2}{(x^{2}+x+1)^{3}}$
Work Step by Step
We can rewrite $h(x)=(x^{2}+x+1)^{-2}$
(h is a composite function)
Let $u(x)=x^{-2},\qquad v(x)=x^{2}+x+1$
Then,$\quad y=h(x)=u(v(x))$ and $h^{\prime}(x)=\displaystyle \frac{dy}{dx}=\frac{du}{dv}\frac{dv}{dx}$
$\displaystyle \frac{du}{dx}=-2x^{-3}, \quad \frac{dv}{dx}=2x+1$
$\displaystyle \frac{du}{dv}=-2v^{-3}=-2(x^{2}+x+1)^{-3}$
$h^{\prime}(x)=\displaystyle \frac{du}{dv}\frac{dv}{dx}=-2(x^{2}+x+1)^{-3}\cdot(2x+1)$
$h^{\prime}(x)=-2(2x+1)(x^{2}+x+1)^{-3}$
$h^{\prime}(x)=-\displaystyle \frac{4x+2}{(x^{2}+x+1)^{3}}$
