To simplify the expression \(\frac{12\sqrt{24}}{3\sqrt{300}}\), we will first simplify each component in the numerator and the denominator.
Starting with \(\sqrt{24}\):
\[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \]
Now substituting this back into the expression:
\[ 12\sqrt{24} = 12 \times 2\sqrt{6} = 24\sqrt{6} \]
Next, we simplify \(\sqrt{300}\):
\[ \sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3} \]
Now substituing this into the expression:
\[ 3\sqrt{300} = 3 \times 10\sqrt{3} = 30\sqrt{3} \]
Now we can rewrite the original expression:
\[ \frac{12\sqrt{24}}{3\sqrt{300}} = \frac{24\sqrt{6}}{30\sqrt{3}} \]
Next, we can simplify the coefficients. The fraction \(\frac{24}{30}\) can be simplified by dividing both the numerator and denominator by 6:
\[ \frac{24}{30} = \frac{4}{5} \]
Now putting it all together, we have:
\[ \frac{24\sqrt{6}}{30\sqrt{3}} = \frac{4}{5} \cdot \frac{\sqrt{6}}{\sqrt{3}} \]
Next, we simplify the radical:
\[ \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{\frac{6}{3}} = \sqrt{2} \]
Thus, we can rewrite the expression as:
\[ \frac{4}{5} \cdot \sqrt{2} = \frac{4\sqrt{2}}{5} \]
Therefore, the quotient in simplest radical form is:
\[ \boxed{\frac{4\sqrt{2}}{5}} \]