Sure, let's find [tex]\( f(3+h) \)[/tex] if [tex]\( f(x) = x^2 + 3x + 5 \)[/tex].
### Step-by-step Solution:
1. Substitute [tex]\( 3 + h \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(3+h) = (3+h)^2 + 3(3+h) + 5
\][/tex]
2. Expand the terms:
[tex]\[
(3+h)^2 = 9 + 6h + h^2
\][/tex]
[tex]\[
3(3+h) = 9 + 3h
\][/tex]
3. Combine all the terms:
[tex]\[
f(3+h) = (9 + 6h + h^2) + (9 + 3h) + 5
\][/tex]
4. Simplify by combining like terms:
[tex]\[
f(3+h) = h^2 + 6h + 3h + 9 + 9 + 5
\][/tex]
[tex]\[
f(3+h) = h^2 + 9h + 23
\][/tex]
Therefore, the correct answer is:
C. [tex]\( h^2 + 9h + 23 \)[/tex]
So, [tex]\( f(3 + h) = h^2 + 9h + 23 \)[/tex].
