Lisa drove 120 miles, then increased her speed by 30 miles per hour for 270 miles. if the second part of the drive took 1 hour longer than the first part, find her average speed

Let the speed = r
120/r = the time it took to go the first 120 hours.

270/(r + 30) = the time it took to go the second part of the trip.
270/(r + 30) - 120/r = 1

Multiply through by r*(r + 30)

270 * r - 120*(r + 30) = 1* r*(r + 30) Remove the brackets.
270r - 120r - 3600 = r^2 + 30r Collect left like terms.
150r - 3600 = r^2 + 30r Transfer the left side to the right.
0 = r^2 - 120r + 3600 This should factor 
(r - 60)(r - 60) = 0
r = 60 and r1 = 60

270 / (30 + 60) = 270 / 90 = 3 hours
120 / 60 = 2 hours. So far so good.

Now we need to get the average speed.
d = 270 + 120
d = 390

t = 3 + 2 
t = 5 hours.

Average speed  = d / t
Average speed  = 390 / 5
Average speed = 78 miles / hour.