(i) Assuming x is the length of the uncut board and y is the breadth of the uncut board,

Area of uncut board - 4 Area of 3x3 squares = 496
xy - 4(3x3) = 496
xy - 36 = 496
xy = 496+36
xy = 532
y = 532/x ————— Eqn 1

Length x Breadth x Height = Volume
(x-3-3)(y-3-3)(3) = 858
(x-6)(y-6)(3) = 858
(xy - 6x - 6y + 36)(3) = 858
xy - 6x - 6y + 36 = 858/3
xy - 6x - 6y + 36 = 286
xy - 6x - 6y + 36 - 286 = 0
xy - 6x - 6y - 250 = 0 (shown) ————— Eqn 2

(ii) Sub Eqn 1 into Eqn 2:
x(532/x) - 6x - 6(532/x) - 250 = 0
532 - 6x - 3192/x - 250 = 0
532x - 6x² - 3192 - 250x = 0
-6x² + 282x - 3192 = 0
x² - 47x + 532 = 0
(x - 19)(x - 28) = 0
x = 19 or 28

y = 532/19 or 532/28
y = 28 or 19

(iii) Length of the box is the longer side.
Therefore length of the box is 28cm.