To determine which expression is equivalent to \( 9x^2 - 25 \) using the distributive property, we can recognize that \( 9x^2 - 25 \) is a difference of squares. The difference of squares can be factored using the formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
In this case, we can write \( 9x^2 - 25 \) as:
\[ (3x)^2 - (5)^2 \]
According to the difference of squares formula, we can factor it as:
\[ (3x - 5)(3x + 5) \]
Now let's match this with the answer choices given:
- \( (3x + 5)(3x - 5) \) - This is correct.
- \( 9x(x - 25) \) - This does not equal \( 9x^2 - 25 \).
- \( (3x + 5)^2 \) - This expands to \( 9x^2 + 30x + 25 \), which is not equal to \( 9x^2 - 25 \).
- \( (4.5x \cdot 12.5)(4.5x - 12.5) \) - This is not in the correct form.
Thus, the expression that is equivalent to \( 9x^2 - 25 \) is:
\[ \boxed{(3x + 5)(3x - 5)} \]