To solve for [tex]\( f(a+h) \)[/tex] where the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to substitute [tex]\( a + h \)[/tex] in place of [tex]\( x \)[/tex] in the given function and then expand and simplify the expression.
Given:
[tex]\[ f(x) = x^2 + 3x + 5 \][/tex]
1. Substitute [tex]\( a + h \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ f(a + h) = (a + h)^2 + 3(a + h) + 5 \][/tex]
2. Expand [tex]\( (a + h)^2 \)[/tex]:
[tex]\[ (a + h)^2 = a^2 + 2ah + h^2 \][/tex]
3. Expand [tex]\( 3(a + h) \)[/tex]:
[tex]\[ 3(a + h) = 3a + 3h \][/tex]
4. Substitute the expanded forms back into the function:
[tex]\[ f(a + h) = a^2 + 2ah + h^2 + 3a + 3h + 5 \][/tex]
So the expanded and simplified expression for [tex]\( f(a + h) \)[/tex] is:
[tex]\[ a^2 + 2ah + h^2 + 3a + 3h + 5 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{a^2 + 2ah + h^2 + 3a + 3h + 5} \][/tex]
This matches option A.
