We need to factor the expression $16m^6 - 25$ completely.

1. Problem Description

We need to factor the expression $16m^6 - 25$ completely.

2. Solution Steps

First, we recognize that the given expression is a difference of squares. We can rewrite it as $(4m^3)^2 - 5^2$.

The difference of squares formula is $a^2 - b^2 = (a - b)(a + b)$. Applying this to our expression, we have:

$16m^6 - 25 = (4m^3)^2 - 5^2 = (4m^3 - 5)(4m^3 + 5)$

Now we need to check if either $(4m^3 - 5)$ or $(4m^3 + 5)$ can be factored further. Since neither of these can be factored further using real numbers, our final factored expression is $(4m^3 - 5)(4m^3 + 5)$.

3. Final Answer

$(4m^3 - 5)(4m^3 + 5)$