To find the value of [tex]\( k \)[/tex] when [tex]\( f(x+3) = x^2 + kx - 21 \)[/tex], we need to substitute [tex]\( x + 3 \)[/tex] into the given function [tex]\( f(x) = x^2 - 2x - 24 \)[/tex] and then compare the resulting expression to [tex]\( x^2 + kx - 21 \)[/tex].
Let's start with the given function:
[tex]\[ f(x) = x^2 - 2x - 24 \][/tex]
We want to find [tex]\( f(x+3) \)[/tex]:
1. Substitute [tex]\( x + 3 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(x+3) = (x + 3)^2 - 2(x + 3) - 24 \][/tex]
2. Expand [tex]\( (x + 3)^2 \)[/tex]:
[tex]\[ (x + 3)^2 = x^2 + 6x + 9 \][/tex]
3. Substitute and simplify:
[tex]\[ f(x+3) = x^2 + 6x + 9 - 2(x + 3) - 24 \][/tex]
[tex]\[ = x^2 + 6x + 9 - 2x - 6 - 24 \][/tex]
[tex]\[ = x^2 + 6x - 2x + 9 - 6 - 24 \][/tex]
[tex]\[ = x^2 + 4x + 9 - 6 - 24 \][/tex]
[tex]\[ = x^2 + 4x - 21 \][/tex]
Now, we compare this expression to [tex]\( x^2 + kx - 21 \)[/tex]:
[tex]\[ x^2 + 4x - 21 \][/tex]
It is clear that the coefficient of [tex]\( x \)[/tex] in [tex]\( x^2 + 4x - 21 \)[/tex] is 4. Hence, [tex]\( k = 4 \)[/tex].
Therefore, the value of [tex]\( k \)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
The correct answer is:
D. 4
