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.