4. Let proposition $P$ : The solution sets of the inequalities $a_{1} x^{2}+b_{1} x+c_{1}>0$ and $a_{2} x^{2}+b_{2} x+c_{2}>0$ with respect to $x$ are the same; Proposition $Q: \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$. Then proposition $Q$
A. is a sufficient and necessary condition for proposition $P$
B. is a sufficient condition for proposition $P$ but not a necessary condition
C. is a necessary condition for proposition $P$ but not a sufficient condition
D. is neither a sufficient condition nor a necessary condition for proposition $P$
Solve:
If $a_{1} x^{2}+b_{1} x+c_{1}>0$ and $a_{2} x^{2}+b_{2} x+c_{2}>0$ have the solution set $\mathbf{R} \nRightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$; if $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}=-1 \nRightarrow$ the solution sets of the inequalities $a_{1} x^{2}+b_{1} x+c_{1}>0$ and $a_{2} x^{2}+b_{2} x+c_{2}>0$ are the same, i.e., proposition $Q$ is neither a sufficient condition nor a necessary condition for proposition $P$. Therefore, the answer is $D$.