The problem involves a circle where the central angle $300^\circ$ is given, and we are tasked to find $x$, which relates to an inscribed angle labeled $2x^\circ$.

Solution:

1. Key Property of Central and Inscribed Angles:
   The measure of an inscribed angle is half the measure of the corresponding central angle that subtends the same arc.
   Thus, in this case: $2x = \frac{300}{2}$

2. Simplify: $2x = 150$

3. Solve for $x$: $x = \frac{150}{2} = 75$

Final Answer:

$x = 75^\circ$

Would you like further details or clarifications?

Here are some related questions you might find helpful:

1. What are central and inscribed angles in a circle?
2. Why is the inscribed angle half the central angle?
3. How do properties of circles relate to this solution?
4. Can this method be applied to non-circular geometries?
5. How would the problem change if the central angle was obtuse?

Tip: Always remember the inscribed angle theorem—it’s very useful in solving circle problems!