3x⁴ - 12x³ - 12x² + 30x

That's missing a constant term so we put + 0
on the end:

3x⁴ - 12x³ - 12x² + 30x + 0

We erase all the letters and exponents

3 -12 -12 30  0

To find f(2) by synthetic substitution:
Put 2 on the left and draw a line:

2 | 3  -12  -12  30  0
  |___________________     
    
Bring down the 3:

2 | 3  -12  -12  30  0
  |___________________
    3

Multiply the 3 on the bottom by the 2 at the left.
Get 6.  Put the 6 above and to the right of the 3,
right under the first -12

2 | 3  -12  -12  30  0
  |      6             
    3

Now combine the -12 and the 6, get -6, and write
the -6 down below the bottom line, right of the 3
and underneath the 6.

2 | 3  -12  -12  30  0
  |      6             
    3   -6

Multiply the -6 on the bottom by the 2 at the left.
Get -12.  Put the -12 above and to the right of the -6,
right under the second -12 on the top row:

2 | 3  -12  -12  30  0
  |      6  -12        
    3   -6

Now combine the two -12's, getting -24, and write
the -24 down below the bottom line, right of the -6
and underneath the -12.

2 | 3  -12  -12  30  0
  |      6  -12        
    3   -6  -24

Multiply the -24 on the bottom by the 2 at the left.
Get -48.  Put the -48 above and to the right of the -24,
right under 30 on the top row:

2 | 3  -12  -12  30  0
  |      6  -12 -48    
    3   -6  -24 

Now combine the 30 and the -48, getting -18, and write
the -18 down below the bottom line, right of the -24
and underneath the -48.

2 | 3  -12  -12  30  0
  |      6  -12 -48    
    3   -6  -24 -18

Multiply the -18 on the bottom by the 2 at the left.
Get -36.  Put the -36 above and to the right of the -18,
right under 0 on the top row:

2 | 3  -12  -12  30   0
  |      6  -12 -48 -36 
    3   -6  -24 -18
 
Finally combine the 0 and the -36, getting -36, and write
the -36 down below the bottom line, right of the -18
and underneath the -36, in the bottom right corner:

2 | 3  -12  -12  30   0
  |      6  -12 -48 -36 
    3   -6  -24 -18 -36

Now it's finished, and f(2) = -36 (the number in the bottom
right corner.

Check synthetic substitution by direct substitution: 
 
3x⁴ - 12x³ - 12x² + 30x

3(2)⁴ - 12(2)³ - 12(2)² + 30(2)

3(16) - 12(8) - 12(4) + 60

48 - 96 - 48 + 60

-48 - 48 + 60

-96 + 60

-36.

So it's correct since direct substitution
gives the same answer.

Now you find f(3) the exact same way.

Edwin