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To solve the problem, let's break it down step by step.

We are given the quadratic function:  
\[ y = x^2 + 2x - 8 \]

### 1. **Finding the x-coordinate of the vertex:**

For a quadratic function in the form \( y = ax^2 + bx + c \), the x-coordinate of the vertex is given by the formula:
\[
x = \frac{-b}{2a}
\]
In our equation, \( a = 1 \) and \( b = 2 \). Substituting these values into the formula:
\[
x = \frac{-2}{2(1)} = -1
\]
Thus, the x-coordinate of the vertex is \( -1 \).

### 2. **Finding the y-coordinate of the vertex:**

Now that we know the x-coordinate of the vertex, we can find the y-coordinate by substituting \( x = -1 \) back into the original equation:
\[
y = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9
\]
Thus, the y-coordinate of the vertex is \( -9 \).

### 3. **Finding the x-intercepts:**

To find the x-intercepts, we set \( y = 0 \) in the equation and solve for \( x \):
\[
0 = x^2 + 2x - 8
\]
We can factor this quadratic equation:
\[
0 = (x + 4)(x - 2)
\]
Setting each factor equal to zero gives the solutions:
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
Thus, the x-intercepts are \( x = -4 \) and \( x = 2 \).

### Final Answer:

- x-coordinate of the vertex: \( -1 \)
- y-coordinate of the vertex: \( -9 \)
- x-intercepts: \( x = -4 \) and \( x = 2 \)

So the correct answer corresponds to option **a**: \( x = -4, x = 2 \).

Would you like more details on any part of the process? Here are 5 related questions:

1. How do you find the vertex of a quadratic function from its equation?
2. What is the geometric significance of the vertex in a parabola?
3. How does factoring help in finding the x-intercepts of a quadratic function?
4. Can you always factor a quadratic equation to find the x-intercepts?
5. What does the discriminant tell you about the number of x-intercepts?

**Tip:** The vertex form of a quadratic equation can be helpful to quickly identify the vertex without using the formula.

Find the x-coordinate of the vertex, y-coordinate of the vertex, and the x-intercepts for the equation y = x^2 + 2x - 8.

Learn how to calculate the vertex and x-intercepts of the quadratic equation y = x^2 + 2x - 8. This detailed step-by-step solution explains the key concepts of graphing parabolas and solving quadratic equations through factoring. Suitable for students in Grades 9-11.

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