Given that the sum of the roots of the quadratic polynomial is $-p$ and the product of the roots is $-\frac{1}{p}$, we can use the standard form of a quadratic equation and Vieta's formulas to find the polynomial.

For a quadratic equation $ax^2 + bx + c = 0$, Vieta's formulas tell us:

• The sum of the roots $\alpha + \beta = -\frac{b}{a}$
• The product of the roots $\alpha \beta = \frac{c}{a}$

Here, we have: $\alpha + \beta = -p$ $\alpha \beta = -\frac{1}{p}$

These match the general form: $-b/a = -p \quad \text{and} \quad c/a = -\frac{1}{p}$

Thus, $b = pa$ and $c = -\frac{a}{p}$.

We can choose $a = k$ (where $k$ is a constant multiplier), then: $b = pk \quad \text{and} \quad c = -\frac{k}{p}$

The quadratic polynomial thus becomes: $kx^2 + (pk)x + \left(-\frac{k}{p}\right)$

Simplifying, we get: $k \left(x^2 + px - \frac{1}{p}\right)$

Therefore, the polynomial $k(-px^2 + \frac{x}{p} + 1)$ seems incorrect based on this reasoning. However, if the given answer is in that form, it may involve some manipulation or context not directly visible from the problem statement alone. The simplest correct form should be:

$k \left(x^2 + px - \frac{1}{p}\right)$

Would you like further details or have any questions?

Here are some related questions you might find useful:

1. How do you derive the quadratic formula using Vieta's formulas?
2. Can you explain the relationship between the roots and coefficients of a quadratic equation?
3. What are some applications of quadratic equations in real life?
4. How do transformations affect the graph of a quadratic function?
5. Can you provide an example of solving a quadratic equation using factoring?

Tip: Always check your polynomial by substituting the roots back to ensure they satisfy the given conditions.