Let's convert the given quadratic equation [tex]\( -2x^2 - 4x + 2y + 8 = 0 \)[/tex] into its vertex form step-by-step.
1. Isolate the [tex]\( y \)[/tex] term:
We want to separate the [tex]\( y \)[/tex] term on one side of the equation. Start by moving the [tex]\( y \)[/tex] and constant terms to the right:
[tex]\[
-2x^2 - 4x + 8 = -2y
\][/tex]
2. Divide by -2 to isolate [tex]\( y \)[/tex]:
To simplify, divide every term by -2:
[tex]\[
x^2 + 2x - 4 = y
\][/tex]
So, the equation is now:
[tex]\[
y = x^2 + 2x - 4
\][/tex]
3. Complete the square:
To express the equation in vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], we'll complete the square for the quadratic expression on the right-hand side.
Start with the quadratic and linear terms:
[tex]\[
x^2 + 2x
\][/tex]
Find the constant term to complete the square. Take half of the coefficient of [tex]\( x \)[/tex], which is 2, and square it:
[tex]\[
\left(\frac{2}{2}\right)^2 = 1
\][/tex]
Add and subtract this square within the equation:
[tex]\[
y = x^2 + 2x + 1 - 1 - 4
\][/tex]
4. Group the perfect square trinomial:
Combine the terms to complete the square within the equation:
[tex]\[
y = (x + 1)^2 - 1 - 4
\][/tex]
5. Simplify:
Combine the constants:
[tex]\[
y = (x + 1)^2 - 5
\][/tex]
6. Identify the vertex form:
The vertex form of the equation is now:
[tex]\[
y = (x + 1)^2 - 5
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{B. \, y = (x + 1)^2 - 5}
\][/tex]
