Is it possible to make screen videos on a unrooted device? The device is Sony Xperia m2 aqua Recently I want to record some screen videos like to show when bugs happen and e.t.c.Is it possible on a unrooted Sony xperia m2 aqua?If yes how?
I see that your device originally came with Android KitKat. You can try a command line method here. screenrecord binary does the screen recording. Its maximum recording duration is 180 seconds. In Kitkat, its usages goes by:

Usage: screenrecord [options] 

Records the device's display to a .mp4 file.

Options:
--size WIDTHxHEIGHT
    Set the video size, e.g. "1280x720".  Default is the device's main
    display resolution (if supported), 1280x720 if not.  For best results,
    use a size supported by the AVC encoder.
--bit-rate RATE
    Set the video bit rate, in megabits per second.  Default 4Mbps.
--time-limit TIME
    Set the maximum recording time, in seconds.  Default / maximum is 180.
--rotate
    Rotate the output 90 degrees.
--verbose
    Display interesting information on stdout.
--help
    Show this message.

In Lollipop, a parameter bugreport is also added:

--bugreport
    Add additional information, such as a timestamp overlay, that is helpful
    in videos captured to illustrate bugs.

An example to record screen using in-built screen recorder (you would need adb setup in PC and USB debugging enabled in the device), 

adb shell screenrecord --bit-rate 8000000  /sdcard/test.mp4


That command will save the screen recording in a file test.mp4 located under /sdcard. Change the bitrate as per your needs, or try an app for GUI.