To find the value of \( \alpha^2 \beta + \alpha \beta^2 \) for the roots \( \alpha \) and \( \beta \) of the quadratic polynomial \( p(x) = 4x^2 - 5x - 1 \), we will utilize relationships from Vieta's formulas and some algebraic manipulation.
Step 1: Identify Roots Using Vieta’s Formulas
- For a quadratic polynomial \( ax^2 + bx + c \):
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots \( \alpha \beta = \frac{c}{a} \)
- Here, for \( p(x) = 4x^2 - 5x - 1 \):
- \( a = 4 \), \( b = -5 \), \( c = -1 \)
- Applying Vieta’s:
- \( \alpha + \beta = \frac{5}{4} \)
- \( \alpha \beta = -\frac{1}{4} \)
Step 2: Calculate \( \alpha^2 \beta + \alpha \beta^2 \)
- We can factor \( \alpha^2 \beta + \alpha \beta^2 \) as:
- \( \alpha^2 \beta + \alpha \beta^2 = \alpha \beta (\alpha + \beta) \)
- Substituting the values from Vieta’s:
- \( \alpha^2 \beta + \alpha \beta^2 = \alpha \beta (\alpha + \beta) \)
- \( = \left(-\frac{1}{4}\right) \left(\frac{5}{4}\right) \)

Step 3: Final Calculation
- Performing the multiplication:
- \( = -\frac{5}{16} \)
Conclusion
- Therefore, the value of \( \alpha^2 \beta + \alpha \beta^2 \) is:
- \( \boxed{-\frac{5}{16}} \)