Example 4. Given the quadratic function $\mathrm{y}=3 \mathrm{p} \mathrm{x}^{2}-6 \mathrm{px} +3 p-2$, for what value of $p$ will the distance between the two intersection points of this parabola with the $x$-axis be $\frac{\sqrt{6}}{3}$.

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The translation maintains the original text's line breaks and formatting.
$$
\begin{array}{l}
\text { Solve } \frac{\sqrt{\triangle}}{|3 p|}=\frac{\sqrt{(-6 p)^{2}-4 \cdot 3 p(3 p-2)}}{|3 p|} \\
=\frac{\sqrt{24 p}}{|3 p|}=\frac{\sqrt{6}}{3} . \\
\end{array}
$$

Solving for $\mathrm{p}$, we get $\mathrm{p}=4$ .