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The expression you are asked to factor is:

\[
4r^2 - 21r + 5
\]

### Step-by-Step Factoring Process:

1. **Identify the form:** This is a quadratic expression in the form \( ax^2 + bx + c \), where \( a = 4 \), \( b = -21 \), and \( c = 5 \).

2. **Find factors of \( a \times c = 4 \times 5 = 20 \)** that add up to \( b = -21 \). The pair of factors that satisfy this condition are \( -20 \) and \( -1 \), since:
   \[
   -20 \times -1 = 20 \quad \text{and} \quad -20 + (-1) = -21
   \]

3. **Rewrite the middle term:** Replace \( -21r \) with \( -20r - r \), so the expression becomes:
   \[
   4r^2 - 20r - r + 5
   \]

4. **Group the terms:**
   \[
   (4r^2 - 20r) - (r - 5)
   \]

5. **Factor each group:**
   \[
   4r(r - 5) - 1(r - 5)
   \]

6. **Factor out the common binomial factor \( (r - 5) \):**
   \[
   (4r - 1)(r - 5)
   \]

Thus, the factored form of the expression is:
\[
(4r - 1)(r - 5)
\]

Would you like further details on any step? Here are some related questions for deeper understanding:

1. How do you determine the factors of a quadratic equation?
2. Can you apply factoring to solve the quadratic equation for \( r \)?
3. What happens if the quadratic is not factorable easily? How can the quadratic formula help?
4. How do you verify that the factorization is correct?
5. How is factoring different from completing the square?

**Tip:** Always check your factored expression by expanding it back out to ensure it matches the original quadratic.

Learn how to factor the quadratic expression 4r^2 - 21r + 5 completely with a detailed step-by-step process. This problem involves factoring by grouping, applying basic algebraic methods suitable for Grades 9-10.

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