To solve this problem, we need to use the properties of angles and arcs in a circle. Let's analyze and solve:

Given:

1. $m\angle AEC = 4x$,
2. $m\overset{\frown}{AC} = 120^\circ$,
3. $m\overset{\frown}{DB} = 2x$.

The angle formed by intersecting chords inside the circle ($\angle AEC$) is related to the intercepted arcs: $m\angle AEC = \frac{1}{2} \left( m\overset{\frown}{AC} + m\overset{\frown}{DB} \right)$

Substituting the known values:

$4x = \frac{1}{2} \left( 120 + 2x \right)$

Simplify:

$4x = \frac{1}{2} \cdot 120 + \frac{1}{2} \cdot 2x$ $4x = 60 + x$

Solve for $x$:

$4x - x = 60$ $3x = 60$ $x = 20$

Final Answer:

The value of $x$ is: $\boxed{20}$

Let me know if you want further clarification!

Related Questions:

1. How are angles formed by intersecting chords related to the intercepted arcs?
2. What is the measure of $\overset{\frown}{DB}$ when $x = 20$?
3. How do we calculate angles when the vertex is inside the circle compared to on the circle?
4. What is the value of $\angle AEC$ in this specific problem?
5. Can this method be used for any type of circle geometry problem?

Tip:

When solving problems involving circle geometry, always check whether angles are inside, on, or outside the circle to apply the correct formulas for their relationship with arcs.