Jane, it's almost 250 miles each way, of which only 30 or so are freeway, and the last hour would be up Provo Canyon. to PC. I don't know what your comfort level with 2 lane mountain roads in Winter in the dark is, but it's beyond mine.
I would recommend against doing this at night in an unfamiliar and unequipped (blankets, chains, coats, gloves, flashlight, etc) rental car.
Remember, it is mostly dry now in the Intermountain West, but you will be between 4200 and over 7000 feet elevations. Weather and road conditions can change rapidly.
We are all hoping for lots of snow soon, so all the visitors (and locals) can go skiing.
Jim