Explanation:

The characteristic equation of a system is given by:

2s^5 + s^4 + 4s^3 + 2s^2 + 2s + 1 = 0

To determine the stability of the system, we need to check the location of the roots of the characteristic equation in the s-plane. There are different approaches to do this, but one common method is to use the Routh-Hurwitz criterion.

Routh-Hurwitz Criterion:

The Routh-Hurwitz criterion is a mathematical tool used to determine the stability of a system by analyzing the location of the roots of its characteristic equation in the complex plane. The criterion states that a system is stable if and only if all the coefficients of the characteristic equation are positive, and all the roots of the equation have negative real parts.

If any of the coefficients are negative, then we can't use the Routh-Hurwitz criterion to determine the stability of a system.

Using the Routh-Hurwitz criterion, we can construct a table that looks like this:

| 2 | 4 | 2 |
| 1 | 2 | 0 |
| 11/4 | 1 | 0 |
| 17/11 | 0 | 0 |
| 2 | 0 | 0 |

The first column of the table contains the coefficients of the even powers of s, and the second column contains the coefficients of the odd powers of s. The remaining entries are obtained by performing some algebraic operations on the coefficients of the first two columns.

To determine the stability of the system, we need to check the sign changes in the first column of the table. In this case, there are two sign changes, which means that there are two roots with positive real parts. Therefore, the system is unstable.

Conclusion:

The answer is option C, i.e., the system is unstable.