The equation you provided is:

\[
x^2 + 6x + 8y + 5 = 0
\]

This represents a parabola, but it's not in its standard form yet. Let's transform it into the standard form of a parabola, \( (x - h)^2 = 4p(y - k) \), by completing the square.

### Steps:
1. Rearrange the terms involving \(x\) on one side:
   \[
   x^2 + 6x = -8y - 5
   \]

2. Complete the square for \(x^2 + 6x\):
   - Take half of the coefficient of \(x\) (which is 6), divide by 2 to get 3, and square it to get 9.
   - Add 9 to both sides:
   \[
   x^2 + 6x + 9 = -8y - 5 + 9
   \]
   This simplifies to:
   \[
   (x + 3)^2 = -8y + 4
   \]

3. Simplify further:
   \[
   (x + 3)^2 = -8(y - \frac{1}{2})
   \]

### Final form:
\[
(x + 3)^2 = -8(y - \frac{1}{2})
\]

This is the equation of a **parabola** that opens **downward** because of the negative coefficient on the \(y\)-term.

### Key features:
- **Vertex**: \( (-3, \frac{1}{2}) \)
- **Axis of symmetry**: The line \(x = -3\)
- **Direction**: The parabola opens downward because the coefficient of \(y\) is negative.
- **Focus and directrix**: These can be determined by the value of \(p\), where \(4p = -8\), so \(p = -2\). The focus is 2 units below the vertex, and the directrix is 2 units above the vertex.

Would you like more details on this? Or do you have any questions?

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Here are 5 questions to explore further:

1. How do you find the focus and directrix of this parabola?
2. What is the significance of completing the square in transforming equations?
3. How would the equation change if the parabola opened upwards?
4. Can you derive the formula for the focus of a parabola in general form?
5. How would this parabola graph compare to a horizontal parabola?

**Tip**: Completing the square is a crucial technique for rewriting quadratic equations in a more useful form, such as for graphing or analyzing the geometry of a parabola.

Explore the transformation of the quadratic equation x² + 6x + 8y + 5 = 0 into the standard form of a parabola. Learn how to complete the square, identify key features like the vertex and direction, and analyze the geometric properties. Ideal for students in Grades 10-12.

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