Let's call the two numbers we're trying to find "x" and "y". We can translate the problem into equations:

The product of two numbers is 4.2 times the sum: xy = 4.2(x + y)

The sum is 5 times its difference: x + y = 5(x - y)

We now have two equations with two unknowns, which we can solve simultaneously:

x + y = 5(x - y) => x + y = 5x - 5y => 6y = 4x => y = (2/3)x

Substitute y in terms of x in the first equation:

xy = 4.2(x + y) => x((2/3)x) = 4.2(x + (2/3)x)

Simplifying the equation:

2x^2/3 = 4.2(5x/3)

2x^2 = 21x

x(2x - 21) = 0

x = 0 or x = 21/2

If x = 0, then y = 0 as well, which means that the product of the two numbers would be 0, not 4.2 times their sum. Therefore, we can discard x = 0 as a solution.

If x = 21/2, then y = (2/3)(21/2) = 7, since y = (2/3)x. We can check that this solution satisfies both equations:

xy = (21/2)(7) = 73.5 4.2(x + y) = 4.2(21/2 + 7) = 73.5 x + y = 21/2 + 7 = 35/2 5(x - y) = 5(21/2 - 7) = 35

Therefore, the two numbers are 21/2 and 7.