The factor theorem says that to find out whether A is a factor of B,
you divide B by A and if you get a zero remainder then A is a factor
of B but if you don't get a zero remainder, then A is not a factor of B.

We want to find out whether (x-3) is a factor of f(x) = x⁴+ 12x³ + 6x + 27,
so we divide x^4+12x^3+6x+27 by (x-3) to see if we get a zero remainder:
         __________________________ 
    x - 3)x⁴+ 12x³ + 0x² + 6x + 27

Notice there was no x² term so we had to insert +0x².
But we have a shortcut way of doing that called "synthetic
division".  We change the sign of the -3 to +3 and write:

       3|1  12   0   6   27
        |    3  45 135  423   
         1  15  45 141  450

So we got 450, not 0. That means that (x-3) is not a factor of
x⁴+ 12x³ + 6x + 27

Edwin