Basically you wish to minimize the area of the cardboard used.
The printed area's length I'll call L
The printed area's width will be 300/L
So, the length of the cardboard will be L + 4 to account for 2 inches at top and bottom.
The width of the cardboard will be 300/L + 2
So the Area of the cardboard A = (L+4)(300/L +2)
A = 300 + 2L + 1200/L + 8
A = 308 + 2L +1200(L)^(-1)
A ' = 2 - 1200/(L^2)
0 = 2 - 1200/(L^2)
2 = 1200/(L^2)
2L^2 = 1200
L^2 = 600
L = sqrt(600)
If you calcualte that to the nearest .001 and then substitute back in to find the width of the printed area, you can find the Dimensions of the cardboard and the area of the cardboard.