The problem asks us to identify the type of curve represented by the polar equation $r = \frac{4}{1 + 2\sin\theta}$. If it is a conic, we must also determine its eccentricity.

1. Problem Description

The problem asks us to identify the type of curve represented by the polar equation $r = \frac{4}{1 + 2\sin\theta}$. If it is a conic, we must also determine its eccentricity.

2. Solution Steps

First, we need to rewrite the polar equation into the standard form of a conic section:

$r = \frac{ed}{1 + e\sin\theta}$

where $e$ is the eccentricity and $d$ is the distance from the focus (the pole) to the directrix.

We have $r = \frac{4}{1 + 2\sin\theta}$. Comparing this to the standard form, we see that $ed = 4$ and $e = 2$. Therefore, $2d = 4$, so $d = 2$.

Since $e = 2 > 1$, the conic section is a hyperbola.

3. Final Answer

The curve is a hyperbola with eccentricity $e = 2$.