To analyze and simplify the given equation \((x - 1)^2 + (2y + 1)^2 + 3 = 8\), let's follow these steps:

1. **Isolate the squared terms:**

\[
(x - 1)^2 + (2y + 1)^2 + 3 = 8
\]

Subtract 3 from both sides:

\[
(x - 1)^2 + (2y + 1)^2 = 5
\]

2. **Recognize the form of the equation:**

The equation \((x - 1)^2 + (2y + 1)^2 = 5\) resembles the standard form of a circle equation \((x - h)^2 + (y - k)^2 = r^2\), but with a modification in the \(y\)-term.

3. **Rewrite the equation to better understand it:**

Let's rewrite the equation in a more familiar form by considering the transformation of the \(y\)-term. Notice that \((2y + 1)^2\) can be expanded, but instead, we can think of it as a scaled circle equation.

\[
(x - 1)^2 + \left(\frac{2y + 1}{\sqrt{5}}\right)^2 = 1
\]

This suggests an ellipse, but let's simplify further by considering the scaling factor.

4. **Simplify the \(y\)-term:**

Let \(u = 2y + 1\). Then the equation becomes:

\[
(x - 1)^2 + u^2 = 5
\]

This is a circle centered at \((1, 0)\) with radius \(\sqrt{5}\) in the \(xu\)-plane.

5. **Transform back to the \(xy\)-plane:**

Recall \(u = 2y + 1\). To understand the shape in the \(xy\)-plane, we need to consider the transformation back:

\[
(x - 1)^2 + (2y + 1)^2 = 5
\]

This represents an ellipse in the \(xy\)-plane, centered at \((1, -\frac{1}{2})\).

6. **Identify the semi-major and semi-minor axes:**

The term \((2y + 1)^2\) suggests a scaling factor along the \(y\)-axis. To find the lengths of the semi-major and semi-minor axes, consider the standard form of an ellipse \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\).

Here, the equation \((x - 1)^2 + (2y + 1)^2 = 5\) can be rewritten as:

\[
\frac{(x - 1)^2}{5} + \frac{(2y + 1)^2}{5} = 1
\]

This simplifies to:

\[
\frac{(x - 1)^2}{5} + \frac{(2y + 1)^2}{5} = 1
\]

Dividing both sides by 5:

\[
\frac{(x - 1)^2}{5} + \frac{(2y + 1)^2}{5} = 1
\]

This confirms that the ellipse is centered at \((1, -\frac{1}{2})\) with semi-major axis \(\sqrt{5}\) along the \(x\)-direction and semi-minor axis \(\frac{\sqrt{5}}{2}\) along the \(y\)-direction.

**Conclusion:**

The given equation \((x - 1)^2 + (2y + 1)^2 + 3 = 8\) represents an ellipse centered at \((1, -\frac{1}{2})\) with semi-major axis \(\sqrt{5}\) and semi-minor axis \(\frac{\sqrt{5}}{2}\).