Use the Distributive Property to find which expression is equivalent to $9x^{2}-25$

To use the Distributive Property on the expression \(9x^2 - 25\), we note that it has the form of a difference of squares. This can be expressed as:

\[
a^2 - b^2 = (a - b)(a + b)
\]

In this case, we can identify:

\[
a = 3x \quad \text{and} \quad b = 5
\]

Thus, we have:

\[
9x^2 - 25 = (3x)^2 - 5^2
\]

Applying the difference of squares formula, we get:

\[
9x^2 - 25 = (3x - 5)(3x + 5)
\]

We compare this result with the provided options:

1. \((3x + 5)^2\) - This expands to \(9x^2 + 30x + 25\), which does not match.
2. \((4.5x + 12.5)(4.5x - 12.5)\) - This is also a difference of squares, but since \(4.5^2\) and \(12.5^2\) do not yield \(9\) and \(25\), this option does not apply.
3. \(9x(x - 25)\) - This expands to \(9x^2 - 225x\), which does not match.
4. \((3x + 5)(3x - 5)\) - This expands to \(9x^2 - 25\), which matches our original expression.

Therefore, the correct expression that is equivalent to \(9x^2 - 25\) is:

\[
\boxed{(3x + 5)(3x - 5)}
\]