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The problem asks for the vertex and the x-intercepts of the graph of the quadratic function \( y = (x - 4)(x + 2) \).

### Step 1: Expand the function
First, let's expand the quadratic function to standard form:
\[
y = (x - 4)(x + 2) 
\]
\[
y = x^2 + 2x - 4x - 8
\]
\[
y = x^2 - 2x - 8
\]

### Step 2: Find the vertex
The vertex form of a quadratic equation is given by:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. But since we have the quadratic in the form \(y = ax^2 + bx + c\), we can find the vertex using the formula:
\[
h = -\frac{b}{2a}
\]
For the equation \( y = x^2 - 2x - 8 \), \( a = 1 \) and \( b = -2 \):
\[
h = -\frac{-2}{2(1)} = \frac{2}{2} = 1
\]
Now, substitute \( x = 1 \) back into the equation to find \( k \):
\[
k = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9
\]
So, the vertex is \( (1, -9) \).

### Step 3: Find the x-intercepts
The x-intercepts occur where \( y = 0 \). So, set the equation to zero:
\[
0 = (x - 4)(x + 2)
\]
Set each factor equal to zero:
\[
x - 4 = 0 \quad \text{or} \quad x + 2 = 0
\]
\[
x = 4 \quad \text{or} \quad x = -2
\]
So, the x-intercepts are \( x = 4 \) and \( x = -2 \).

### Conclusion:
- The vertex is \( (1, -9) \) (Option A).
- The x-intercepts are \( 4 \) and \( -2 \), which correspond to the points \( (4, 0) \) and \( (-2, 0) \) (Option D).

Therefore, the correct answers are:
- Vertex: **A. (1, -9)**
- x-intercepts: **D. (-4, 0), (-2, 0)**

Would you like any further details on this problem? 

Here are 5 related questions to expand on this concept:
1. How do you convert a quadratic equation from standard form to vertex form?
2. What is the significance of the discriminant in determining the nature of the roots?
3. How can the axis of symmetry of a parabola be found directly from the standard form of the quadratic equation?
4. What is the role of the coefficient \(a\) in the graph of the quadratic equation \(y = ax^2 + bx + c\)?
5. How would the vertex and intercepts change if the equation was \(y = - (x - 4)(x + 2)\)?

**Tip:** For any quadratic function, the x-intercepts can be quickly found by setting the equation to zero and solving for \(x\), and the vertex can be found using the vertex formula.

What are the vertex and x-intercepts of the graph of y = (x - 4)(x + 2)? Select one answer for the vertex and one for the x-intercepts.

Learn how to determine the vertex and x-intercepts of the quadratic function y = (x - 4)(x + 2). This problem includes key algebraic methods such as expanding quadratics and using the vertex formula. Suitable for students in Grade 9-10.

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