Let x be the length of the printed material, in centimeters.
Then the wide is  centimeters.

Accounting for the margins, the area of poster is

S = (x + 2*6)*(384/x + 2*6),  or

S = (x+12)*(384/x + 12),  or

S = .   (3)

The problem asks to minimize S as a function of "x".

To do it, take the derivative  of (3).

 = -12*(384/x^2) + 12.

The condition  = 0 leads to the equation

 = 1,   or

 = .

Then x =  = 19.6 cm is the solution (approximately).


Answer.  The poster's dimensions providing minimum area with the given text area and given margins 

         are 19.6+12 = 31.6 and  centimeters.