Both roots of the polynomial s2+4Ks+(K−3)𝑠2+4𝐾𝑠+(𝐾−3), where K𝐾 is a real valued parameter, would have a negative real part for allK>0𝐾>0K<3𝐾<3K>3𝐾>3K<4

The roots of the polynomial s^2 + 4Ks + (K-3) are given by the quadratic formula:

s = [-4K ± sqrt((4K)^2 - 41(K-3))]/(2*1) = [-4K ± sqrt(16K^2 - 4K + 12)]/2 = [-4K ± sqrt(16K^2 - 4K + 12)]/2

For the roots to have a negative real part, the term under the square root (the discriminant) must be positive (since the square root of a negative number would result in an imaginary part), and -4K must be negative.

The discriminant is 16K^2 - 4K + 12. For this to be positive:

16K^2 - 4K + 12 > 0

This is a quadratic inequality. To solve it, we first find the roots of the corresponding quadratic equation:

16K^2 - 4K + 12 = 0

Using the quadratic formula, the roots are:

K = [4 ± sqrt((-4)^2 - 41612)]/(2*16) = [4 ± sqrt(16 - 768)]/32 = [4 ± sqrt(-752)]/32

Since the term under the square root is negative, there are no real roots. Therefore, the inequality 16K^2 - 4K + 12 > 0 holds for all real K.

However, we also need -4K to be negative, which means K must be positive. Therefore, the roots of the polynomial s^2 + 4Ks + (K-3) will have a negative real part for all K > 0.