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The given function is \( f(x) = x^2 + 3x + 5 \), and you are asked to find \( f(3 + h) \).

To solve this, we need to substitute \( x = 3 + h \) into the function:

\[
f(3 + h) = (3 + h)^2 + 3(3 + h) + 5
\]

Let's expand and simplify this step by step:

1. Expand \( (3 + h)^2 \):
   \[
   (3 + h)^2 = 9 + 6h + h^2
   \]

2. Expand \( 3(3 + h) \):
   \[
   3(3 + h) = 9 + 3h
   \]

3. Add all the terms together:
   \[
   f(3 + h) = (9 + 6h + h^2) + (9 + 3h) + 5
   \]

4. Combine like terms:
   \[
   f(3 + h) = h^2 + (6h + 3h) + (9 + 9 + 5)
   \]
   \[
   f(3 + h) = h^2 + 9h + 23
   \]

Thus, \( f(3 + h) = h^2 + 9h + 23 \).

Would you like further details or have any other questions? 

Here are some related questions to deepen your understanding:
1. How do we handle similar problems if we substitute a different expression for \( x \)?
2. Can you explain the significance of combining like terms during the simplification?
3. What if we substituted \( x = -h \)? How would the solution change?
4. What are some real-world applications of quadratic functions like this one?
5. Can you graph \( f(x) \) and \( f(3 + h) \) and observe the changes visually?

**Tip:** Whenever you're dealing with function substitution, keep track of each term in the expression and simplify step by step to avoid errors.

Given f(x) = x^2 + 3x + 5, find f(3 + h).

Learn how to solve function substitution problems like f(3 + h) with detailed steps. This example includes quadratic expansion and combining like terms, suitable for Grades 8-10.

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