To find [tex]\( f(3 + h) \)[/tex] for the given function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to substitute [tex]\( x = 3 + h \)[/tex] into the function.
1. Start with the function definition:
[tex]\[
f(x) = x^2 + 3x + 5
\][/tex]
2. Substitute [tex]\( x = 3 + h \)[/tex] into the function:
[tex]\[
f(3 + h) = (3 + h)^2 + 3(3 + h) + 5
\][/tex]
3. Expand [tex]\( (3 + h)^2 \)[/tex]:
[tex]\[
(3 + h)^2 = 9 + 6h + h^2
\][/tex]
4. Expand [tex]\( 3(3 + h) \)[/tex]:
[tex]\[
3(3 + h) = 9 + 3h
\][/tex]
5. Substitute these expanded terms back into the function:
[tex]\[
f(3 + h) = 9 + 6h + h^2 + 9 + 3h + 5
\][/tex]
6. Combine like terms:
[tex]\[
f(3 + h) = h^2 + 9h + 23
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{h^2 + 9h + 23}
\][/tex]
Thus, the correct option is:
[tex]\[
\text{C}.\ h^2 + 9 h + 23
\][/tex]
