+22 Find the range of the function
\(h(x)=\frac{5x^2+20x+33}{x^2+4x+7}\)
Enter your answer in interval notation.
+22 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 \(\boxed{h(x)\in[\frac{13}{3},5)}\)
