Interesting Range Question; Can use polynomial long division to solve

Ok, I found the answer.

We can use polynomial long division to simplify: 

$\frac{5x^2+20x+33}{x^2+4x+7}=5-\frac{2}{x^2+4x+7}$

Then, h(x)<5, and we can find the minimum by completing the square on the quadratic.

$x^2+4x+7=(x+2)^2+3$,

so the minimum is 3. Hence, the least possible value of h(x) is $\frac{13}{3}$, and the range is $\boxed{h(x)\in[\frac{13}{3},5)}$