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.