What does Descartes Rule of Signs tell you about the real roots of the polynomial

-2x^3+3x^3-5x-2=0
possible answers
one positive root either 2 or 0 neg
2 or 0 pos and 2 or 0 neg
one pos and one neg
either 2 or 0 pos and one neg

Please explain

To apply Descartes Rule of Signs to determine the number of positive and negative roots of a given polynomial equation, we need to follow these steps:

1. Rearrange the given equation in descending order of powers and make sure all the terms are in one side of the equation, equal to zero.
- In this case, we have: -2x^3 + 3x^2 - 5x - 2 = 0

2. Count the number of sign changes in the coefficients when going from left to right.
- In our equation, the coefficients are -2, 3, -5, -2. There are two sign changes: from -2 to 3 and then from -5 to -2.

3. The number of positive roots is either equal to the number of sign changes or less than that by an even number.
- In our case, the number of sign changes is 2, so the number of positive roots could be 2 or 0.

4. Replace "x" with "-x" in the equation and count the number of sign changes in the coefficients when going from left to right.
- By replacing "x" with "-x" in our equation, we get: -2(-x)^3 + 3(-x)^2 - 5(-x) - 2 = 0
Simplifying it: 2x^3 + 3x^2 + 5x - 2 = 0
- Counting the sign changes in the coefficients, we have one sign change: from 3 to 5.

5. The number of negative roots is either equal to the number of sign changes or less than that by an even number.
- In our case, the number of sign changes is 1, so the number of negative roots could be 1 or 0.

Based on Descartes Rule of Signs, the possible answers for the real roots of the polynomial -2x^3 + 3x^2 - 5x - 2 = 0 are:
- One positive root and one negative root (one pos and one neg).
- This is because the number of positive roots could be 2 or 0, and the number of negative roots could be 1 or 0.