For 

*f*(*x*) = *x*2 + 6*x* + 9,

 evaluate 

*f*(*x* + 4) − *f*(*x*).

 Simplify.

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.

Learn how to evaluate and simplify f(x + 4) - f(x) for the quadratic function f(x) = x^2 + 6x + 9. Step-by-step solution with detailed algebraic manipulation and simplification. Suitable for high school students studying quadratic equations.

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