A bird leaves its nest and travels 20 miles per hour downwind for x hours. On the return trip, the bird travels 4 miles per hour slower and has 6 miles left after x hours. What is the distance of the entire trip? How long does the entire trip take?

We need to calculate a total TIME.  
Let the time required to cover distance d at 20 mph be x.  d is the ONE WAY distance, not the total distance traveled (which would be 2d).

Remember the formula d=rt:  the distance traveled equals the rate times the time.  Here, d = rx, and thus x = d/r  = d/(20mph)

Let's begin by recognizing  that       d(going) = d(returning)
Then                                            (20 mph)x   = 6 + (16 mph)x

                                                      (20 mph)x - (16 mph)x = 6 miles
Then:                                                    (4 mph)x = 6 miles
or:                                                             x = 1.5 hours

It takes the bird 1.5 hours at 20 mph to cover distance d, which is 30 miles.
Thus, the total time spent flying is 1.5 hours (going) and [1.5 hours + (6 miles)/(16 mph) returning:

                            1.5 hours + 1.5 hours + 0.375 hours = 3.375 hours total.

The total distance covered is 2d, or 2(30 miles) = 60 miles.

It took me a while and a lot of experimentation to arrive at these results.  Please, if my arguments here are not clear, ask questions.