for the function f(x)=4$$ x^2 $$$$ +4x-2^{} $$. evaluate and simplify f(x+h)=

We are given the function:  

\[
f(x) = 4x^2 + 4x - 2
\]

We need to evaluate $ f(x+h) $, which means substituting $ x+h $ for $ x $ in the function.

### Step 1: Substitute $ x+h $ into $ f(x) $

\[
f(x+h) = 4(x+h)^2 + 4(x+h) - 2
\]

### Step 2: Expand the Squared Term  

\[
(x+h)^2 = x^2 + 2xh + h^2
\]

\[
4(x+h)^2 = 4(x^2 + 2xh + h^2) = 4x^2 + 8xh + 4h^2
\]

### Step 3: Expand the Linear Term  

\[
4(x+h) = 4x + 4h
\]

### Step 4: Combine All Terms  

\[
f(x+h) = 4x^2 + 8xh + 4h^2 + 4x + 4h - 2
\]

This is the simplified expression for $ f(x+h) $.

---

Would you like me to expand further or clarify any part of the process?  

Here are some related questions to deepen understanding:  

1. How do we compute the difference quotient $\frac{f(x+h) - f(x)}{h}$ for this function?  
2. What is the derivative $ f'(x) $ using the definition of a derivative?  
3. How does substituting $ h = 0 $ in $ f(x+h) $ verify $ f(x) $?  
4. How can we use $ f(x+h) $ to approximate values of $ f(x) $ for small $ h $?  
5. What happens to $ f(x+h) $ as $ h \to 0 $, and why is this important in calculus?  

### **Tip:**  
When expanding $ (x+h)^2 $, always remember the middle term $ 2xh $, as it's commonly missed in algebraic expansions!

for the function f(x)=4x^2 + 4x - 2, evaluate and simplify f(x+h)

Learn how to simplify f(x+h) for the function f(x) = 4x^2 + 4x - 2, step-by-step. This problem demonstrates function transformation and polynomial expansion, suitable for students in Grades 9-11.

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