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We are given the quadratic function \( f(x) = 8x + 4x^2 - 2 \), and the question asks for the vertex of the function. 

To find the vertex of a quadratic function \( f(x) = ax^2 + bx + c \), we can use the formula for the x-coordinate of the vertex:
\[
x = \frac{-b}{2a}
\]
where \( a \) is the coefficient of \( x^2 \) and \( b \) is the coefficient of \( x \).

### Step 1: Identify the coefficients
For the given function \( f(x) = 4x^2 + 8x - 2 \), we have:
- \( a = 4 \)
- \( b = 8 \)
- \( c = -2 \) (though \( c \) is not needed to find the vertex)

### Step 2: Apply the formula for the x-coordinate of the vertex
\[
x = \frac{-8}{2 \times 4} = \frac{-8}{8} = -1
\]

### Step 3: Find the y-coordinate by substituting \( x = -1 \) back into the function
\[
f(-1) = 4(-1)^2 + 8(-1) - 2
\]
\[
f(-1) = 4(1) - 8 - 2 = 4 - 8 - 2 = -6
\]

### Conclusion:
The vertex of the quadratic function is \( (-1, -6) \).

So, the correct answer is **\((-1, -6)\)**.

Let me know if you want more details on this. Here are some additional related questions:

1. How do you find the axis of symmetry from a quadratic function?
2. Can you explain the significance of the vertex in a parabola?
3. How can you rewrite the given quadratic function in vertex form?
4. What is the general shape of a parabola for a quadratic function with a positive \(a\)?
5. How would the graph change if the quadratic function was \( -4x^2 + 8x - 2 \)?

**Tip:** The vertex form of a quadratic function is useful when you want to easily identify the vertex and the direction in which the parabola opens.

Which is the vertex of the quadratic function f(x) = 8x + 4x^2 - 2?

Discover how to determine the vertex of the quadratic function f(x) = 8x + 4x^2 - 2 with a step-by-step solution. This example demonstrates key algebraic concepts and formulas, including the vertex formula, suitable for students in Grades 8-10.

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