1. **State the problem:** We need to find the value of $x$ given that the angle at point $D$ inside the circle is $4x$ and the angle at point $E$ on the circumference is $62^\circ$. 2. **Recall the circle theorem:** The angle at the center (or inside the circle) is twice the angle at the circumference subtended by the same chord. This means: $$4x = 2 \times 62^\circ$$ 3. **Set up the equation:** $$4x = 124^\circ$$ 4. **Solve for $x$:** $$x = \frac{124^\circ}{4}$$ 5. **Simplify the fraction:** $$x = \frac{\cancel{124}^\circ}{\cancel{4}} = 31^\circ$$ 6. **Final answer:** $$x = 31^\circ$$