The given expression is:

\[ 40r^3 + 200r^2 - 5r - 25 \]

To factor this expression, let's follow these steps:

1. **Factor out the greatest common factor (GCF):**
   The GCF of the terms is 5. Factoring out 5, we get:

   \[
   5(8r^3 + 40r^2 - r - 5)
   \]

2. **Factor the remaining cubic polynomial (if possible):**
   To factor \( 8r^3 + 40r^2 - r - 5 \), we can use the method of grouping:

   Group the terms:

   \[
   (8r^3 + 40r^2) - (r + 5)
   \]

   Factor out the common factors from each group:

   \[
   8r^2(r + 5) - 1(r + 5)
   \]

   Notice that both groups have a common factor of \( (r + 5) \):

   \[
   (8r^2 - 1)(r + 5)
   \]

3. **Factor further (if possible):**
   The expression \( 8r^2 - 1 \) is a difference of squares, so it can be factored:

   \[
   8r^2 - 1 = (2\sqrt{2}r - 1)(2\sqrt{2}r + 1)
   \]

   So, the complete factorization is:

   \[
   5(2\sqrt{2}r - 1)(2\sqrt{2}r + 1)(r + 5)
   \]

Thus, the fully factored form of the expression is:

\[
\boxed{5(2\sqrt{2}r - 1)(2\sqrt{2}r + 1)(r + 5)}
\]

Would you like more details on any step or have any questions?

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Here are five related questions you might consider:

1. How do you determine the greatest common factor (GCF) in a polynomial?
2. What is the method of grouping, and when is it useful in factoring?
3. How do you recognize and factor a difference of squares?
4. Can you factor a cubic polynomial using synthetic division?
5. What are some other common methods for factoring polynomials?

**Tip:** Always check for common factors before attempting more complex factoring methods.

Learn how to factor the polynomial 40r^3 + 200r^2 - 5r - 25 with a detailed step-by-step solution. This problem covers polynomial factoring techniques, including the grouping method and recognizing a difference of squares. Suitable for advanced high school students studying algebra.

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