The function given is:

$f(x) = 2x^2 + 8x + 3$

To determine the vertex form, we need to complete the square.

Step 1: Factor out the leading coefficient of $x^2$

$f(x) = 2(x^2 + 4x) + 3$

Step 2: Complete the square inside the parentheses

We take half of the coefficient of $x$ (which is 4), divide by 2 to get 2, and then square it to get $2^2 = 4$.

So we add and subtract 4 inside the parentheses: $f(x) = 2(x^2 + 4x + 4 - 4) + 3$ $f(x) = 2((x + 2)^2 - 4) + 3$

Step 3: Simplify the expression

$f(x) = 2(x + 2)^2 - 8 + 3$ $f(x) = 2(x + 2)^2 - 5$

Thus, the vertex form of the function is: $f(x) = 2(x + 2)^2 - 5$

Step 4: Determine the minimum or maximum value

Since the coefficient of $(x + 2)^2$ is positive (2), the parabola opens upwards, indicating a minimum value. The minimum value occurs at the vertex, where $x = -2$, and the function value is $f(-2) = -5$.

Correct Answer:

The vertex form is $f(x) = 2(x + 2)^2 - 5$, and the minimum value is $-5$. Therefore, the correct answer is D.

Would you like further details on any step?

Here are five related questions you might explore:

1. How do you derive the vertex form from any quadratic equation?
2. What is the significance of the vertex in a quadratic function?
3. How do you find the axis of symmetry for a parabola?
4. How does the coefficient of the quadratic term affect the direction of the parabola?
5. What are the applications of vertex form in real-world problems?

Tip: Completing the square is a useful method to rewrite a quadratic equation in vertex form, which makes it easier to find the maximum or minimum value of the function.