Find the eccentricity, foci and the length of the latusrectum of the ellipse `x^2+4y^2+8y-2x+1=0.`

Explanation

To solve the given ellipse equation \( x^2 + 4y^2 + 8y - 2x + 1 = 0 \) and find the eccentricity, foci, and the length of the latus rectum, we will follow these steps:

### Step 1: Rearranging the Equation
First, we rearrange the equation to group the \( x \) and \( y \) terms together:
\[ x^2 - 2x + 4y^2 + 8y + 1 = 0 \]

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