Given the `alpha + beta` are the roots of the quadratic equation `px^(2) + qx + 1 = 0` find the value of `alpha^(3)beta^(2) + alpha^(2)beta^(3)`.

Answer

To solve the problem, we need to find the value of \( \alpha^3 \beta^2 + \alpha^2 \beta^3 \) given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( px^2 + qx + 1 = 0 \).

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is \( px^2 + qx + 1 = 0 \). Here, we identify:
- \( a = p \)
- \( b = q \)
- \( c = 1 \)

2. **Use Vieta's Formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{q}{p} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{1}{p} \)

3. **Rewrite the expression**:
We need to find \( \alpha^3 \beta^2 + \alpha^2 \beta^3 \). This can be factored as:
\[ \alpha^3 \beta^2 + \alpha^2 \beta^3 = \alpha^2 \beta^2 (\alpha + \beta) \]

4. **Substitute the values**:
We already found:
- \( \alpha + \beta = -\frac{q}{p} \)
- \( \alpha \beta = \frac{1}{p} \)

Therefore, \( \alpha^2 \beta^2 = (\alpha \beta)^2 = \left(\frac{1}{p}\right)^2 = \frac{1}{p^2} \).

5. **Combine the results**:
Now substitute these values into the expression:
\[ \alpha^3 \beta^2 + \alpha^2 \beta^3 = \alpha^2 \beta^2 (\alpha + \beta) = \frac{1}{p^2} \left(-\frac{q}{p}\right) \]
Simplifying this gives:
\[ = -\frac{q}{p^3} \]

### Final Answer:
Thus, the value of \( \alpha^3 \beta^2 + \alpha^2 \beta^3 \) is:
\[ -\frac{q}{p^3} \]