4. The parabola $y=a x^{2}+b x+c(a>0)$ intersects the $y$-axis at $P$, and intersects the $x$-axis at different points $A, B, |O A|=\frac{|O B|}{2}=\frac{|O P|}{3}$. Then the product of all possible values of $b$ is ( ).
(A) $\frac{729}{16}$
(B) $-\frac{9}{4}$
(C) $-\frac{9}{2}$
(D) $\frac{81}{64}$
4.(A).
(1) When $A$ and $B$ are both on the left side of the $y$-axis,
$$
\because a>0, \Delta>0 \text {, }
$$
$\therefore$ point $P$ is above the $x$-axis.
Let $A(-k, 0)$, $B(-2k, 0)$, $P(0, 3k)$. Substituting these three points into $y = ax^2 + bx + c$, we get $b = \frac{9}{2}$.
(2) When $A$ and $B$ are both on the right side of the $y$-axis, similarly, we get $b = -\frac{9}{2}$.
(3) When $A$ and $B$ are on opposite sides of the $y$-axis (note that $|AO| = \frac{|BO|}{2}$), point $P$ is below the $x$-axis. By analogy with (1) and (2), we get $b = \frac{3}{2}$ or $-\frac{3}{2}$.

Thus, $\frac{9}{12} \times \left(-\frac{9}{2}\right) \times \frac{3}{2} \times \left(-\frac{3}{2}\right) = \frac{729}{16}$.