If 
f(x) = 8x2 − x + 8
 and h ≠ 0, find the following and simplify.
(a)    
f(x + h)
 

To find \( f(x + h) \) given that \( f(x) = 8x^2 - x + 8 \), we will substitute \( x + h \) into the function \( f(x) \) in place of \( x \).

### Step-by-Step Solution:
1. **Substitute \( x + h \) into \( f(x) \):**

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

2. **Expand \( (x + h)^2 \):**

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

   Therefore,

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

3. **Distribute the 8 across the terms inside the parentheses:**

   \[
   f(x + h) = 8x^2 + 16xh + 8h^2 - x - h + 8
   \]

4. **Combine like terms:**

   Since there are no further like terms to combine, the expression simplifies to:

   \[
   f(x + h) = 8x^2 + 16xh + 8h^2 - x - h + 8
   \]

Thus, the simplified form of \( f(x + h) \) is:

\[
\boxed{8x^2 + 16xh + 8h^2 - x - h + 8}
\]

Would you like to explore more details or have any further questions?

### Relative Questions:
1. How would you find \( f(x) + f(h) \) for the same function?
2. What is the value of \( f'(x) \) using the derivative of the given function?
3. How does \( f(x+h) - f(x) \) simplify in this case?
4. What is the difference between the original function and its derivative?
5. How do the terms involving \( h \) disappear when finding the derivative using the definition?

### Tip:
When substituting expressions like \( x + h \) into a function, always ensure you carefully expand and simplify each term to avoid any mistakes.

Learn how to find f(x + h) for the function f(x) = 8x^2 - x + 8. Follow a detailed step-by-step solution to expand and simplify the expression. Suitable for advanced high school students.

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