The given network function is (3s^2 + 8s) / [(s + 1)(s + 3)]. This represents an LC impedance. Let's understand why.

Explanation:
An impedance is a measure of opposition to the flow of an alternating current (AC) in a circuit. It consists of both resistance (R) and reactance (X). Reactance can be either inductive (XL) or capacitive (XC).

In the given network function, we have a polynomial in the numerator (3s^2 + 8s) and a product of two polynomials in the denominator (s + 1)(s + 3). By analyzing the form of the network function, we can determine the type of impedance it represents.

1. Numerator Analysis:
The numerator (3s^2 + 8s) represents a second-degree polynomial in 's'. This indicates that the impedance has a reactive component. A second-degree polynomial suggests the presence of an inductive or capacitive reactance.

2. Denominator Analysis:
The denominator is a product of two first-degree polynomials (s + 1)(s + 3). This suggests that there are two poles in the impedance function. Each pole corresponds to a reactive component.

3. Pole Analysis:
The poles of the impedance function are the roots of the denominator polynomials. For the given function, the poles are -1 and -3 (obtained by setting s + 1 and s + 3 equal to zero).

Now, let's analyze the poles:
- The pole at s = -1 indicates a reactive component of (s + 1), which corresponds to an inductive impedance (XL).
- The pole at s = -3 indicates a reactive component of (s + 3), which corresponds to a capacitive impedance (XC).

Since the given network function has both inductive and capacitive reactances, it represents an LC impedance.

Conclusion:
The network function (3s^2 + 8s) / [(s + 1)(s + 3)] represents an LC impedance because it has both inductive and capacitive reactances in its numerator and denominator.