Given the function $f(x) = x^2 + 6x + 9$, we need to evaluate $f(x + 4) - f(x)$ and simplify the expression.

Step 1: Find $f(x + 4)$

Substitute $x + 4$ into the function $f(x) = x^2 + 6x + 9$: $f(x + 4) = (x + 4)^2 + 6(x + 4) + 9$

Step 2: Expand $(x + 4)^2$ and $6(x + 4)$

$(x + 4)^2 = x^2 + 8x + 16$ $6(x + 4) = 6x + 24$

Step 3: Combine all terms for $f(x + 4)$

$f(x + 4) = x^2 + 8x + 16 + 6x + 24 + 9$ $f(x + 4) = x^2 + 14x + 49$

Step 4: Find $f(x + 4) - f(x)$

Now subtract $f(x)$ from $f(x + 4)$: $f(x + 4) - f(x) = (x^2 + 14x + 49) - (x^2 + 6x + 9)$

Step 5: Simplify the expression

$f(x + 4) - f(x) = x^2 + 14x + 49 - x^2 - 6x - 9$ $f(x + 4) - f(x) = (x^2 - x^2) + (14x - 6x) + (49 - 9)$ $f(x + 4) - f(x) = 8x + 40$

The simplified expression for $f(x + 4) - f(x)$ is:

$f(x + 4) - f(x) = 8x + 40$

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

1. How would the process change if the function were cubic instead of quadratic?
2. What is the significance of finding the difference $f(x + a) - f(x)$ in calculus?
3. Can you calculate $f(x + 2) - f(x)$ for the given function?
4. What happens if you compute $f(x - 4) - f(x)$?
5. How do you factor the expression $f(x + 4) - f(x)$ further?

Tip: When evaluating expressions like $f(x + a) - f(x)$, carefully expand and simplify each step to avoid mistakes.