To solve the equation [tex]\( 2x^2 + 16x - 8 = 0 \)[/tex] using the method of completing the square, follow these steps:
1. Divide the whole equation by 2 to simplify it:
[tex]\[
\frac{2x^2 + 16x - 8}{2} = 0 \implies x^2 + 8x - 4 = 0
\][/tex]
2. Rearrange the equation to set it in the form where the linear and quadratic terms are grouped together:
[tex]\[
x^2 + 8x = 4
\][/tex]
3. Complete the square by adding and subtracting the square of half the coefficient of [tex]\( x \)[/tex]. Here, the coefficient of [tex]\( x \)[/tex] is 8, so [tex]\(\left( \frac{8}{2} \right)^2 = 16\)[/tex]:
[tex]\[
x^2 + 8x + 16 - 16 = 4
\][/tex]
Which simplifies to:
[tex]\[
(x + 4)^2 - 16 = 4
\][/tex]
4. Isolate the perfect square term by adding 16 to both sides:
[tex]\[
(x + 4)^2 = 20
\][/tex]
5. Solve for [tex]\( x \)[/tex] by taking the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[
x + 4 = \pm \sqrt{20}
\][/tex]
Since [tex]\( \sqrt{20} \)[/tex] can be simplified to [tex]\( 2\sqrt{5} \)[/tex]:
[tex]\[
x + 4 = \pm 2\sqrt{5}
\][/tex]
6. Solve for [tex]\( x \)[/tex] by subtracting 4 from both sides:
[tex]\[
x = -4 \pm 2\sqrt{5}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x = -4 \pm 2\sqrt{5}}
\][/tex]
So, the correct option is:
[tex]\[
\boxed{D. \, x = -4 \pm 2\sqrt{5}}
\][/tex]
