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The given expression is:

\[
5x^2 + 29x + 20
\]

To factor this quadratic expression completely, we need to find two numbers that multiply to \(5 \times 20 = 100\) and add up to \(29\).

1. **Identify the coefficients**:
   - \(a = 5\)
   - \(b = 29\)
   - \(c = 20\)

2. **Find factors of \(100\) that add up to \(29\)**:
   - The numbers \(25\) and \(4\) satisfy this condition because \(25 + 4 = 29\) and \(25 \times 4 = 100\).

3. **Rewrite the middle term**:
   We can rewrite \(29x\) as \(25x + 4x\), so the expression becomes:
   \[
   5x^2 + 25x + 4x + 20
   \]

4. **Factor by grouping**:
   Group terms as follows:
   \[
   (5x^2 + 25x) + (4x + 20)
   \]
   Factor out the common factors from each group:
   \[
   5x(x + 5) + 4(x + 5)
   \]

5. **Factor out the common binomial \((x + 5)\)**:
   \[
   (5x + 4)(x + 5)
   \]

So, the completely factored form of the expression is:

\[
(5x + 4)(x + 5)
\]

Let me know if you would like more details or have questions!

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Here are five related questions for further practice:
1. How would you factor \(6x^2 + 11x + 4\)?
2. What are the roots of the equation \(5x^2 + 29x + 20 = 0\)?
3. How can you verify that \((5x + 4)(x + 5)\) is the correct factorization?
4. How would you factor a trinomial with a negative constant term?
5. What methods can be used to factor trinomials with larger coefficients?

**Tip:** When factoring, always check if there is a greatest common factor (GCF) that can be factored out first to simplify the expression!

Learn to factor the quadratic expression 5x^2 + 29x + 20 completely with step-by-step instructions. This solution explores algebraic concepts, including factoring by grouping and identifying suitable factors for efficient problem-solving. Ideal for Grades 8-10 students.

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