Sure, let's solve this problem step-by-step together.
We have two functions defined as follows:
1. [tex]\( g(x) = -x^2 - 5x + 1 \)[/tex]
2. [tex]\( h(x) = (x + 3)^2 - 11 \)[/tex]
To find [tex]\( g(h(-5)) \)[/tex], we first need to evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = -5 \)[/tex].
Step 1: Evaluate [tex]\( h(-5) \)[/tex]
[tex]\[ h(x) = (x + 3)^2 - 11 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\(-5\)[/tex]:
[tex]\[ h(-5) = (-5 + 3)^2 - 11 \][/tex]
[tex]\[ h(-5) = (-2)^2 - 11 \][/tex]
[tex]\[ h(-5) = 4 - 11 \][/tex]
[tex]\[ h(-5) = -7 \][/tex]
So, [tex]\( h(-5) = -7 \)[/tex].
Step 2: Evaluate [tex]\( g(h(-5)) \)[/tex], which is essentially evaluating [tex]\( g(-7) \)[/tex]
We now have [tex]\( h(-5) = -7 \)[/tex]. So, next we need to find [tex]\( g(-7) \)[/tex]:
[tex]\[ g(x) = -x^2 - 5x + 1 \][/tex]
Replace [tex]\( x \)[/tex] with [tex]\(-7\)[/tex]:
[tex]\[ g(-7) = -(-7)^2 - 5(-7) + 1 \][/tex]
[tex]\[ g(-7) = -49 + 35 + 1 \][/tex]
[tex]\[ g(-7) = -49 + 36 \][/tex]
[tex]\[ g(-7) = -13 \][/tex]
Therefore, [tex]\( g(h(-5)) = g(-7) = -13 \)[/tex].
So the final answer is [tex]\(-13\)[/tex].
