Explanation:

Let's break down the problem:

- Let the two natural numbers be x and y, with x > y.

- The square of their difference: (x - y)² = 324 ? x - y = 18 (since 18² = 324; we take positive root for natural numbers).

- Their product: x·y = 144.

- We are asked for the positive difference between the squares: |x² - y²|.

Calculation:

- x² - y² = (x - y)(x + y) = 18·(x + y).

- From earlier: x·y = 144 and x - y = 18.

- So, x and y are roots of t² - (x + y)t + x·y = 0.

- Let S = x + y.

- x and y: roots of t² - St + 144 = 0, with x - y = 18.

- System:

- x + y = S

- x - y = 18

- Solving:

- x = (S + 18)/2

- y = (S - 18)/2

- x·y = 144 ? [(S + 18)/2]·[(S - 18)/2] = 144 ? (S² - 324)/4 = 144 ? S² - 324 = 576 ? S² = 900 ? S = 30

- So, x + y = 30.

- Therefore, x² - y² = 18 × 30 = 540

Let’s match with options:

- Option 1: 630 ?

- Option 2: 540

- Option 3: 450 ?

- Option 4: 360 ?

Key Points:

- The correct answer is Option 2: 540. 

- The method: Find x - y, use product xy, solve for x + y, then find the difference of their squares.

- Other options don’t match the computed result.