Let's assume that the two natural numbers are x and y, where x is the smaller number.
From the problem, we know that:
x + y = 45 ... (1) (The sum of the two numbers is 45)
xy = 4x^2 ... (2) (The product of the two numbers is four times the square of the lesser number)
We can rearrange equation (1) to get:
y = 45 - x
Substituting this into equation (2), we get:
x(45 - x) = 4x^2
Expanding the left side and simplifying, we get:
45x - x^2 = 4x^2
Rearranging and simplifying further, we get a quadratic equation:
5x^2 - 45x = 0
We can factor out x to get:
x(5x - 45) = 0
So, either x = 0 (which is not a natural number) or 5x - 45 = 0.
Solving for x, we get:
5x - 45 = 0
5x = 45
x = 9
So the smaller number is 9, and using equation (1), we can find that the larger number is:
y = 45 - x = 45 - 9 = 36
Therefore, the two natural numbers are 9 and 36.