The total of arcs around a circle is 360°. Thus, the length of the arc DE can be found by adding the given values and choosing the measure that makes the total match 360°.
... 20° + 30° + 40° + DE + 110° + 90° = 360°
... DE = 360° - 290° = 70°
The arc of interest is the sum
... BE = BC + CD + DE
... BE = 30° + 40° + 70°
... BE = 140°