Sure, let's analyze the given quadratic function [tex]\( f(x) = -2(x + 6)(x + 4) \)[/tex].
### Step 1: Expand the Function
First, expand [tex]\( f(x) \)[/tex] to write it in standard quadratic form [tex]\( ax^2 + bx + c \)[/tex]:
[tex]\[
f(x) = -2(x + 6)(x + 4)
\][/tex]
Using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x + 6)(x + 4) = x^2 + 4x + 6x + 24 = x^2 + 10x + 24
\][/tex]
So,
[tex]\[
f(x) = -2(x^2 + 10x + 24) = -2x^2 - 20x - 48
\][/tex]
### Step 2: Determine the Vertex
The standard form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex]. In our case, [tex]\( a = -2 \)[/tex], [tex]\( b = -20 \)[/tex], and [tex]\( c = -48 \)[/tex].
For a quadratic function [tex]\( ax^2 + bx + c \)[/tex], the x-coordinate of the vertex (maximum or minimum point) is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Plugging in our values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x = -\frac{-20}{2 \cdot -2} = \frac{20}{-4} = -5
\][/tex]
### Step 3: Determine the Nature of the Vertex
A quadratic function [tex]\( ax^2 + bx + c \)[/tex] has a:
- Maximum if [tex]\( a < 0 \)[/tex]
- Minimum if [tex]\( a > 0 \)[/tex]
Since [tex]\( a = -2 \)[/tex] in our function and it is negative, the function has a maximum.
### Step 4: Calculate the Maximum Value
To find the maximum value of the function, we need to evaluate [tex]\( f(x) \)[/tex] at the vertex:
[tex]\[
f(-5) = -2(-5 + 6)(-5 + 4)
\][/tex]
Simplify within the parentheses:
[tex]\[
f(-5) = -2(1)(-1) = -2 \cdot -1 = 2
\][/tex]
### Conclusion
The quadratic function [tex]\( f(x) = -2(x + 6)(x + 4) \)[/tex] has a maximum value, and the maximum value is [tex]\( \boxed{2} \)[/tex].
