To rewrite the given quadratic function $F(x) = -2x^2 - 8x + 15$ in vertex form, we follow the process of completing the square.

Steps to Convert to Vertex Form:

1. Factor out the coefficient of $x^2$ from the first two terms: $F(x) = -2(x^2 + 4x) + 15$

2. Complete the square inside the parentheses:

   • Take half the coefficient of $x$, square it, and add it inside the parentheses.
   • The coefficient of $x$ is $4$, so $\left(\frac{4}{2}\right)^2 = 4$.

   Add and subtract $4$ inside the parentheses: $F(x) = -2(x^2 + 4x + 4 - 4) + 15$

3. Factor the perfect square trinomial and simplify:

   • $x^2 + 4x + 4$ factors as $(x + 2)^2$.
   • Simplify: $F(x) = -2((x + 2)^2 - 4) + 15$

4. Distribute $-2$ and combine constants: $F(x) = -2(x + 2)^2 + 8 + 15$ $F(x) = -2(x + 2)^2 + 23$

Final Vertex Form:

$F(x) = -2(x + 2)^2 + 23$

Vertex:

The vertex is $(-2, 23)$.

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Do you want further details or have any questions? Here are 5 related questions to deepen understanding:

1. How do you find the axis of symmetry from the vertex form?
2. What role does the coefficient of $(x + 2)^2$ play in the graph?
3. Can you graph the function using the vertex and another point?
4. How does the negative sign in front of $2$ affect the shape of the parabola?
5. How can this process be used in real-world applications?

Tip: Completing the square is a valuable skill for solving quadratic equations and understanding their properties!