The open loop transfer function of a unity gain negative feedback system is given by,

G(s)=K/s²+4s-5

The range of K for which the system is stable, is

The correct option is K > 5.

To determine the range of $K$ for which the system is stable, we begin by finding the closed-loop transfer function of the system. The system is described as a unity-gain negative feedback system with the open-loop transfer function:

$G\left(s\right)=\frac{K}{s^2+4s-5}$

For a unity feedback system, the feedback path transfer function is $H\left(s\right)=1$. The characteristic equation of the closed-loop system is given by:

$1+G\left(s\right)H\left(s\right)=0$

Substituting $G\left(s\right)$ and $H\left(s\right)=1$ into the equation, we get:

$1+\frac{K}{s^2+4s-5}=0$

Multiplying the entire equation by the denominator $s^2+4s-5$, we obtain the simplified characteristic equation:

$s^2+4s-5+K=0$

Grouping the constant terms together yields:

$s^2+4s+(K-5)=0$

For a second-order system represented by the characteristic equation $s^2+a_{1}s+a_0=0$ to be stable, all the coefficients of the polynomial must be strictly positive (greater than zero). Thus, we establish the following conditions:

1. The coefficient of $s$ is $4$, which is positive ($4>0$). This condition is satisfied.
2. The constant term must also be positive:

$K-5>0$

Solving this inequality for $K$, we find:

$K>5$

Therefore, the closed-loop system is stable when the gain $K$ is strictly greater than 5.