If
f(x) = x2 + 6x + 3,
what is
f(x + 3)?
(Simplify your answer completely.)
f(x + 3) =

To find \( f(x + 3) \), we need to substitute \( x + 3 \) into the function \( f(x) = x^2 + 6x + 3 \).

1. Start with the function:  
   \[
   f(x) = x^2 + 6x + 3
   \]
   
2. Replace \( x \) with \( x + 3 \):  
   \[
   f(x + 3) = (x + 3)^2 + 6(x + 3) + 3
   \]
   
3. Expand \( (x + 3)^2 \):  
   \[
   (x + 3)^2 = x^2 + 6x + 9
   \]
   
4. Distribute the 6 in \( 6(x + 3) \):  
   \[
   6(x + 3) = 6x + 18
   \]
   
5. Substitute these into the function:  
   \[
   f(x + 3) = (x^2 + 6x + 9) + (6x + 18) + 3
   \]

6. Combine like terms:  
   \[
   f(x + 3) = x^2 + 6x + 6x + 9 + 18 + 3
   \]
   \[
   f(x + 3) = x^2 + 12x + 30
   \]

So, the simplified expression for \( f(x + 3) \) is:

\[
f(x + 3) = x^2 + 12x + 30
\]

Would you like further details or have any questions? Here are some related questions:

1. What is the general process for finding \( f(x + a) \) for any given \( f(x) \)?
2. How do you expand a binomial like \( (x + a)^2 \)?
3. Can you provide an example where \( a \) is a different number, say \( f(x + 2) \)?
4. How would you find the derivative of \( f(x) = x^2 + 6x + 3 \)?
5. What would \( f(x - 3) \) be for this function?

**Tip:** When substituting expressions into a function, always expand and combine like terms carefully to simplify your final answer.

Learn how to simplify f(x + 3) for the function f(x) = x^2 + 6x + 3. Follow a step-by-step process to substitute and expand, including key algebraic concepts such as function substitution and expanding binomials. Suitable for students in Grades 9-12.

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