Your answer is almost perfecly OK with me, with one minor problem:
so weight of carbon (Mr 12) would be 44/12 or roughly 3.7.
Calculations are OK, but your explanation seems to be misleading. 3.7 is a coefficient to calculate mass of carbon dioxide from mass of burnt carbon.
Other than that IMHO your approach is perfectly correct. Probably assuming that driving your kid to school you drowe 50 mph for 25 minutes is a little bit stretching the reality, but who knows
However, my professor was talking about how the units were wrong.
You have not listed units in many places, so they have to be guessed - that's wrong. But it doesn't mean units as such are wrong. They are all the time OK.
miles should be into kilometers, which I dont understand why you would do that.
And you are right not understanding - there is no need for that. That's just thinking inside of the box. You know the distance in miles, you know mpg (thats in miles) converting to kilometers is additional steps that will just make calcualtion longer, completely not necessary. And in the final result miles cancel out.
And that the density of gasoline (because you need the mass) is will be required to find an answer.
You have used the density - even if not directly - after all 6.4 pound per gallon IS a density unit. Not metric, but perfectly fit for the data you use, as you are given efficiency in mpg, so using gallon as a volume unit is OK (just like using miles was OK), in the end it cancels out.
Mass of the gasoline can be calculated other way around - from the volume and density, but if you know mass of the gallon from the reliable source, looking for density just to make your calculations longer is absurd.
Is what I did the right thing?
Abso-bloody-lutely