Answer :
To solve this problem, let's carefully analyze the function \( f(x) = (x + 2)(x + 6) \).
### Step 1: Find the Roots of the Function
The roots of \( f(x) \) are the values of \( x \) that make \( f(x) \) equal to zero. We set the function equal to zero and solve for \( x \):
[tex]\[ (x + 2)(x + 6) = 0 \][/tex]
This equation will be zero if either \( x + 2 = 0 \) or \( x + 6 = 0 \). Thus, the roots are:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = -6 \][/tex]
These roots divide the number line into three intervals:
1. \( (-\infty, -6) \)
2. \( (-6, -2) \)
3. \( (-2, \infty) \)
### Step 2: Determine the Sign of \( f(x) \) in Each Interval
#### Interval: \( (-\infty, -6) \)
Choose a test point in this interval. For example, \( x = -7 \):
[tex]\[ f(-7) = (-7 + 2)(-7 + 6) = (-5)(-1) = 5 \][/tex]
Since the value is positive, \( f(x) \) is positive in this interval.
#### Interval: \( (-6, -2) \)
Choose a test point in this interval. For example, \( x = -4 \):
[tex]\[ f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4 \][/tex]
Since the value is negative, \( f(x) \) is negative in this interval.
#### Interval: \( (-2, \infty) \)
Choose a test point in this interval. For example, \( x = 0 \):
[tex]\[ f(0) = (0 + 2)(0 + 6) = (2)(6) = 12 \][/tex]
Since the value is positive, \( f(x) \) is positive in this interval.
### Step 3: Summarize the Findings
Based on the above intervals and the corresponding signs of \( f(x) \), we have:
- The function \( f(x) \) is positive in the intervals \( (-\infty, -6) \) and \( (-2, \infty) \).
- The function \( f(x) \) is negative in the interval \( (-6, -2) \).
### Conclusion
By analyzing all the intervals, we can conclude that the correct statement is:
The function is negative for all real values of \( x \) where \( -6 < x < -2 \).
Therefore, the correct answer is:
The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].
### Step 1: Find the Roots of the Function
The roots of \( f(x) \) are the values of \( x \) that make \( f(x) \) equal to zero. We set the function equal to zero and solve for \( x \):
[tex]\[ (x + 2)(x + 6) = 0 \][/tex]
This equation will be zero if either \( x + 2 = 0 \) or \( x + 6 = 0 \). Thus, the roots are:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = -6 \][/tex]
These roots divide the number line into three intervals:
1. \( (-\infty, -6) \)
2. \( (-6, -2) \)
3. \( (-2, \infty) \)
### Step 2: Determine the Sign of \( f(x) \) in Each Interval
#### Interval: \( (-\infty, -6) \)
Choose a test point in this interval. For example, \( x = -7 \):
[tex]\[ f(-7) = (-7 + 2)(-7 + 6) = (-5)(-1) = 5 \][/tex]
Since the value is positive, \( f(x) \) is positive in this interval.
#### Interval: \( (-6, -2) \)
Choose a test point in this interval. For example, \( x = -4 \):
[tex]\[ f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4 \][/tex]
Since the value is negative, \( f(x) \) is negative in this interval.
#### Interval: \( (-2, \infty) \)
Choose a test point in this interval. For example, \( x = 0 \):
[tex]\[ f(0) = (0 + 2)(0 + 6) = (2)(6) = 12 \][/tex]
Since the value is positive, \( f(x) \) is positive in this interval.
### Step 3: Summarize the Findings
Based on the above intervals and the corresponding signs of \( f(x) \), we have:
- The function \( f(x) \) is positive in the intervals \( (-\infty, -6) \) and \( (-2, \infty) \).
- The function \( f(x) \) is negative in the interval \( (-6, -2) \).
### Conclusion
By analyzing all the intervals, we can conclude that the correct statement is:
The function is negative for all real values of \( x \) where \( -6 < x < -2 \).
Therefore, the correct answer is:
The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].
