How do I replicate these measures in MuseScore?

2 Voices.
Beam Over Rest.
Hide the unwanted rests.

If you have deleted the rests in voice 2 put them back by swapping voice 1 and voice 2 and back again (Tools>Voices)

Then:
- in the selection filter (enable it from the view menu if it is not already visible) deselect voice 2,
- select the range of notes in voice 1 that you want to beam together
- in the Beam properties palette select Join beams.
- repeat for each group of notes that you want beamed together.
- back to the selection filter and reselect voice 2 and deselect voice 1
- repeat the beaming exercise for voice 2.
- When done reselect voice 1 in the selection filter.

At this stage you will still have rests visible.

• select the whole of the section you have just adjusted the beaming in
• right click on one rest and in the pop up menu use Select all similar in this range.
• You can then make the rests invisible.
• You could if you really want delete the rests in voice 2 but not those in voice 1 and leaving them invisible is useful if you want to go back and edit it again.

Sample attached.