i dont fully understand what you mean by "the product of the two numbers and their difference is largest."

do you want the biggest product or the biggest difference? which one?

biggest difference is obvious: 8 and 0

they have to be positive numbers:

a + b = 8 ---> b = (8 - a)

let z = ab, then z = a*(8-a). dz/da = -2a + 8. there are extrema when dz/da = 0. so set:

0 = -2a + 8 ----> a = 4 -----> b = 4

thus a = 4 and b = 4