To rewrite the quadratic function [tex]\(h(x) = -3x^2 - 6x + 5\)[/tex] in vertex form, we need to complete the square. The vertex form of a quadratic function is given by [tex]\(h(x) = a(x - h)^2 + k\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Let's go through the steps to convert the given equation.
### Step 1: Factor out the coefficient of [tex]\(x^2\)[/tex]
First, we factor out the coefficient of [tex]\(x^2\)[/tex] (which is [tex]\(-3\)[/tex]) from the terms involving [tex]\(x\)[/tex]:
[tex]\[h(x) = -3(x^2 + 2x) + 5.\][/tex]
### Step 2: Complete the square
Take the coefficient of [tex]\(x\)[/tex] from inside the parentheses (which is [tex]\(2\)[/tex]), divide by 2, and square it:
[tex]\[
\left(\frac{2}{2}\right)^2 = 1.
\][/tex]
Now add and subtract this square inside the parentheses:
[tex]\[
h(x) = -3(x^2 + 2x + 1 - 1) + 5.
\][/tex]
This can be rewritten by grouping the perfect square trinomial and then subtracting the [tex]\(1\)[/tex] we added:
[tex]\[
h(x) = -3((x^2 + 2x + 1) - 1) + 5.
\][/tex]
Simplify the group inside the parentheses:
[tex]\[
h(x) = -3((x + 1)^2 - 1) + 5.
\][/tex]
### Step 3: Distribute and combine like terms
Now distribute the [tex]\(-3\)[/tex] and simplify:
[tex]\[
h(x) = -3(x + 1)^2 + 3 + 5.
\][/tex]
Combine the constants:
[tex]\[
h(x) = -3(x + 1)^2 + 8.
\][/tex]
So, the vertex form of the given quadratic function is:
[tex]\[
h(x) = -3(x + 1)^2 + 8.
\][/tex]
Looking at the multiple-choice options, the correct answer is:
[tex]\[
h(x) = -3(x + 1)^2 + 8.
\][/tex]
Therefore, the right choice is:
[tex]\[
\boxed{h(x) = -3(x+1)^2+8}
\][/tex]
