To determine which functions have a zero at [tex]\( x = -2 \)[/tex], we need to evaluate each function at [tex]\( x = -2 \)[/tex] and check if the value is zero. Here's the step-by-step solution:
1. Function [tex]\( f(x) = x^2 + 6x + 8 \)[/tex]
Evaluate at [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = (-2)^2 + 6(-2) + 8 = 4 - 12 + 8 = 0
\][/tex]
Since [tex]\( f(-2) = 0 \)[/tex], [tex]\( f(x) \)[/tex] has a zero at [tex]\( x = -2 \)[/tex].
2. Function [tex]\( g(x) = (x + 7)(x - 2) \)[/tex]
Evaluate at [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = (-2 + 7)(-2 - 2) = 5 \cdot (-4) = -20 \neq 0
\][/tex]
Since [tex]\( g(-2) \neq 0 \)[/tex], [tex]\( g(x) \)[/tex] does not have a zero at [tex]\( x = -2 \)[/tex].
3. Function [tex]\( h(x) = 3(x + 2) \)[/tex]
Evaluate at [tex]\( x = -2 \)[/tex]:
[tex]\[
h(-2) = 3(-2 + 2) = 3 \cdot 0 = 0
\][/tex]
Since [tex]\( h(-2) = 0 \)[/tex], [tex]\( h(x) \)[/tex] has a zero at [tex]\( x = -2 \)[/tex].
4. Function [tex]\( k(x) = 4x^2 + 11x + 6 \)[/tex]
Evaluate at [tex]\( x = -2 \)[/tex]:
[tex]\[
k(-2) = 4(-2)^2 + 11(-2) + 6 = 4 \cdot 4 - 22 + 6 = 16 - 22 + 6 = 0
\][/tex]
Since [tex]\( k(-2) = 0 \)[/tex], [tex]\( k(x) \)[/tex] has a zero at [tex]\( x = -2 \)[/tex].
5. Function [tex]\( p(x) = 5x - 10 \)[/tex]
Evaluate at [tex]\( x = -2 \)[/tex]:
[tex]\[
p(-2) = 5(-2) - 10 = -10 - 10 = -20 \neq 0
\][/tex]
Since [tex]\( p(-2) \neq 0 \)[/tex], [tex]\( p(x) \)[/tex] does not have a zero at [tex]\( x = -2 \)[/tex].
Based on the evaluations above, the functions that have a zero at [tex]\( x = -2 \)[/tex] are [tex]\( f(x) \)[/tex], [tex]\( h(x) \)[/tex], and [tex]\( k(x) \)[/tex].
Therefore, the correct answer to the question is:
H. I, III, and IV
So, the selected choice is:
c. H
