Sure! Let's solve the given problem step-by-step.
We are given the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] and we are asked to find [tex]\( f(3+h) \)[/tex].
1. Substitute [tex]\( x \)[/tex] with [tex]\( 3+h \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(3+h) = (3+h)^2 + 3(3+h) + 5
\][/tex]
2. Expand the squared term [tex]\((3+h)^2\)[/tex]:
[tex]\[
(3+h)^2 = 3^2 + 2 \cdot 3 \cdot h + h^2 = 9 + 6h + h^2
\][/tex]
3. Distribute and simplify the other term [tex]\( 3(3+h) \)[/tex]:
[tex]\[
3(3+h) = 3 \cdot 3 + 3 \cdot h = 9 + 3h
\][/tex]
4. Substitute back and combine all terms:
[tex]\[
f(3+h) = 9 + 6h + h^2 + 9 + 3h + 5
\][/tex]
5. Add up all like terms:
[tex]\[
h^2 + (6h + 3h) + (9 + 9 + 5) = h^2 + 9h + 23
\][/tex]
Thus, the function [tex]\( f(3+h) \)[/tex] simplifies to:
[tex]\[
f(3+h) = h^2 + 9h + 23
\][/tex]
As per the choices given, the correct answer is:
A. [tex]\( h^2 + 9h + 23 \)[/tex]
