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To find the equation of the parabola in the form \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex and \(p\) represents the focal length, let's analyze the given information.

### Step-by-Step Solution:

1. **Identify the Vertex and Another Point:**
   - The vertex is \((-4, 3)\).
   - Another point on the parabola is \((4, 5)\).

2. **Use the Vertex Form for a Sideways Parabola:**
   - Since the equation is \((y - k)^2 = 4p(x - h)\), it suggests a parabola that opens horizontally.
   - The vertex form here is centered at \((-4, 3)\), so \(h = -4\) and \(k = 3\).

3. **Plug in the Other Point to Find \(p\):**
   - Substitute \( (x, y) = (4, 5) \) into the equation \((y - k)^2 = 4p(x - h)\).
   - This gives us:
     \[
     (5 - 3)^2 = 4p(4 + 4)
     \]
     Simplifying, we get:
     \[
     2^2 = 4p \cdot 8
     \]
     \[
     4 = 32p
     \]
     \[
     p = \frac{4}{32} = \frac{1}{8}
     \]

4. **Write the Equation of the Parabola:**
   - Now that we have \(h = -4\), \(k = 3\), and \(p = \frac{1}{8}\), the equation becomes:
     \[
     (y - 3)^2 = 4 \cdot \frac{1}{8}(x + 4)
     \]
     Simplifying, we get:
     \[
     (y - 3)^2 = \frac{1}{2}(x + 4)
     \]

### Final Answer:
The equation of the parabola is:
\[
(y - 3)^2 = \frac{1}{2}(x + 4)
\]

Would you like further details or have any questions?

Here are five related questions to explore:

1. How can we determine the direction of a parabola based on its equation?
2. What is the significance of the parameter \(p\) in a parabola's equation?
3. How does changing \(p\) affect the shape of the parabola?
4. What are the steps to convert the equation of a parabola to the standard form?
5. How would the equation change if the vertex were located at a different point?

**Tip:** Remember that for parabolas that open horizontally, the squared term is on the \(y\) variable rather than \(x\), and \(p\) determines the width and direction of the parabola’s opening.

Write the equation of a parabola with its vertex at (-4, 3) and another point (4, 5), in the form (y - k)^2 = 4p(x - h).

Learn how to derive the equation of a parabola with a vertex at (-4, 3) and passing through the point (4, 5), presented in the vertex form (y - k)^2 = 4p(x - h). This problem covers key concepts in coordinate geometry and is suitable for students in Grades 10-12.

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