The given equation:

\[
\frac{(x+2)^2}{16} + \frac{(y-1)^2}{25} = 1
\]

represents the equation of an ellipse in standard form. Here's how we interpret it:

1. **Center of the ellipse**: The equation is in the form \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where \((h, k)\) is the center of the ellipse. Comparing with your equation, the center is at \((-2, 1)\).
  
2. **Semi-major and semi-minor axes**:
   - \(a^2 = 16\) so \(a = 4\), which represents the distance from the center to the vertices along the **x-axis**.
   - \(b^2 = 25\) so \(b = 5\), which represents the distance from the center to the vertices along the **y-axis**.

3. **Orientation**: Since \(b > a\), the ellipse is vertically oriented, meaning the major axis is along the **y-axis**.

4. **Vertices**: The vertices are located along the major axis. For a vertical ellipse:
   - The vertices on the **y-axis** are at \((h, k \pm b)\), which gives \((-2, 1 + 5)\) and \((-2, 1 - 5)\), or \((-2, 6)\) and \((-2, -4)\).
   - The co-vertices on the **x-axis** are at \((h \pm a, k)\), which gives \((-2 + 4, 1)\) and \((-2 - 4, 1)\), or \((2, 1)\) and \((-6, 1)\).

5. **Foci**: The foci are calculated using the formula \(c^2 = b^2 - a^2\). Thus,
   \[
   c^2 = 25 - 16 = 9 \quad \Rightarrow \quad c = 3
   \]
   The foci are along the **y-axis**, located at \((h, k \pm c)\), which gives the foci at \((-2, 1 + 3)\) and \((-2, 1 - 3)\), or \((-2, 4)\) and \((-2, -2)\).

### Summary:
- Center: \((-2, 1)\)
- Semi-major axis: \(b = 5\) (vertical orientation)
- Semi-minor axis: \(a = 4\)
- Vertices: \((-2, 6)\) and \((-2, -4)\)
- Co-vertices: \((2, 1)\) and \((-6, 1)\)
- Foci: \((-2, 4)\) and \((-2, -2)\)

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Would you like more details on any part of this? Here are some related questions to deepen your understanding:

1. How do we derive the standard form of an ellipse equation from general form?
2. What are the real-world applications of ellipses?
3. How can we graph an ellipse given its equation?
4. What are eccentricity and directrices of an ellipse, and how are they calculated?
5. How does the equation of a circle relate to the equation of an ellipse?

**Tip**: In any ellipse, the sum of the distances from any point on the ellipse to the two foci is always constant.

Explore the solution to the ellipse equation (x+2)²/16 + (y-1)²/25 = 1, including key concepts like center, vertices, semi-major and semi-minor axes, and foci. Suitable for students in Grades 10-12 studying conic sections and coordinate geometry.

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