Cubic polynomial unable to be factored nicely

[anniecvc]

Found this on a test for an integrated algebra 2 high school math class!

Factor completely.

6x³ - 3x² + 12

The Attempt at a Solution

3( 2x³ - x² + 4) eq.1

At this point I checked for rational roots using the rational roots theorem and synthetically dividing. I got nothing.

Although this doesn't get us anywhere x² can be factored out of the first two terms:

3[ x²(2x-1) +4)] eq. 2

I then plugged this guy into Wolfram and it is giving me 1 nasty irrational and 2 nasty complex roots. I'm convinced this is NOT "nicely" factorable. Does anyone have any insights?

 

[Mentallic]

You're correct that it can't be factorized using the rational root theorem, so as such, I wouldn't go any further with it. Since it was in a past exam, I'm sure it was a typo that they ended up correcting for the students on the day.