cycle of moon's motion relative to stars compare to cycle of moon's motion to the sun?

Its very simple. The time it takes the moon to complete an orbit relative to the sun is longer compared to the orbit relative to the stars. It takes the moon an extra degree to be parallel to the sun to the sun again.

The Moon returns to the same position relative to the stars every 27.3 days (the Sidereal Month). The Moon returns to the same position relative to the Sun every 29.5 days (the Synodic Month). These periods vary slightly, but only by a few hours. Option 4 looks like a misprint because it's saying the same thing twice, but it's not correct to say 'at times' star cycles are shorter because star cycles are 'always' shorter. So the answer is 2.

the moon path around the sun is tied to the earths orbit around the sun... its view of the stars changes the same as the earth as they both orbit the sun in the same manner... if u are only talking about the cycles and position of the moon and earth cycles... and when they match up... they change a little each month...

Moon goes round the earth once in 27days 8hours with respect to stars.It goes round the Sun along with earth takes one year to go round the sun!

you'll have to study it, b/c there isnt the amount of time im willing to commit to tell you to explain it all