Solution:
The given characteristic equation is s⁴ + s³ + 2s² + 2s + 3 = 0.

To determine the number of roots in the RHS of the s-plane, we need to find the roots of the equation.

We can use numerical methods such as the Routh-Hurwitz criteria or graphical methods such as the root locus plot to find the roots. However, in this case, we can apply the Descartes' rule of signs to determine the number of positive and negative roots.

According to Descartes' rule of signs, the number of positive roots of an equation is equal to the number of sign changes in its coefficients or less by a multiple of 2. Similarly, the number of negative roots is equal to the number of sign changes in the coefficients of the equation when alternate coefficients are changed in sign or less by a multiple of 2.

In this case, the coefficients of the equation are {1, 1, 2, 2, 3}. There are no sign changes in the coefficients of the equation. Therefore, there are no positive roots in the s-plane.

To determine the number of negative roots, we need to change alternate coefficients in sign and find the number of sign changes.

If we change alternate coefficients in sign, we get the equation s⁴ - s³ + 2s² - 2s + 3 = 0.

The coefficients of this equation are {1, -1, 2, -2, 3}. There are two sign changes in the coefficients of this equation. Therefore, there are two negative roots in the s-plane.

Hence, the given characteristic equation s⁴ + s³ + 2s² + 2s + 3 = 0 has two roots in the RHS of the s-plane.

Therefore, the correct answer is option (c) two roots in the RHS of the s-plane.