9. (16 points) Let the function
$$
f(x)=3 a x^{2}-2 b x+c \text {. }
$$

If $a-b+c=0, f(0)>0, f(1)>0$, find the range of $\frac{a+3 b+7 c}{2 a+b}$.
II. 9. Note that, $f(0)=c>0$,
$$
\begin{array}{l}
f(1)=3 a-2 b+c>0, \\
a-b+c=0 .
\end{array}
$$

From equations (1) and (3), we get
$$
a-b>0 \Rightarrow 2 a>b \text {. }
$$

From equations (4) and (5), we get $2 a>b>a$.
Substituting equation (3) into equation (2) yields $a>c \Rightarrow a>0$.
From equation (5), we get $1<\frac{b}{a}<2$, then let
$$
\begin{aligned}
y & =\frac{a+3 b+7 c}{2 a+b}=\frac{10 b-6 a}{2 a+b} \\
& =10-\frac{26}{\frac{b}{a}+2} \in\left(\frac{4}{3}, \frac{7}{2}\right) .
\end{aligned}
$$