Asheet of cardboard 10 inches by 12 inches will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. if the area of the base is to be 80 square inches, then what size square should be cut from each corner?

The attached figure represents the cardboard (10inches by 12 inches) and the squares that should be cut to make the box.

let the length of the square = x
So , the length of the box = L = 12 - 2x
       the width of the box = W = 10 - 2x

And, the area = L * W = 80           ⇒(given)
∴   L * W = (12-2x)(10-2x) = 80

∴ (12 - 2x)(10-2x) =80               
    4x² - 44x + 120 = 80        ⇒ multiply the brackets     
    4x² - 44x +120 - 80 = 0    ⇒ make all variables in one side
    4x² - 44x + 40 = 0            ⇒ sum the similar
    x² - 11x +10 = 0               ⇒ solve by analysis
    (x-1)(x-10) = 0
∴     x = 10 (rejected because the cardboard length = 10 inch)
OR  x = 1
  ∴   the size of the square should be cut from each corner = 1 inch by 1 inch