Find and simplify each of the following for ​f(x)equals2 x squared minus 5 x plus 8.
​(A) f left parenthesis x plus h right parenthesis
​(B) f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis
​(C) StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction
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Part 1
​(A) f left parenthesis x plus h right parenthesisequals
  
enter your response here ​(Do not​ factor.) 

Let's solve the problem step by step.

### Given:
The function is \( f(x) = 2x^2 - 5x + 8 \).

### Part (A) \( f(x + h) \):

To find \( f(x + h) \), substitute \( x + h \) for \( x \) in the function \( f(x) \):

\[
f(x + h) = 2(x + h)^2 - 5(x + h) + 8
\]

Now, expand the expression:

1. Expand \( (x + h)^2 \):
   \[
   (x + h)^2 = x^2 + 2xh + h^2
   \]

2. Substitute this into the function:
   \[
   f(x + h) = 2(x^2 + 2xh + h^2) - 5(x + h) + 8
   \]

3. Distribute the constants:
   \[
   f(x + h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 8
   \]

So, the simplified expression for \( f(x + h) \) is:

\[
\boxed{f(x + h) = 2x^2 + 4xh + 2h^2 - 5x - 5h + 8}
\]

### Part (B) \( f(x + h) - f(x) \):

Next, subtract \( f(x) \) from \( f(x + h) \):

\[
f(x + h) - f(x) = \left(2x^2 + 4xh + 2h^2 - 5x - 5h + 8\right) - \left(2x^2 - 5x + 8\right)
\]

Now, simplify the expression:

1. Combine like terms:
   \[
   f(x + h) - f(x) = 4xh + 2h^2 - 5h
   \]

So, the simplified expression for \( f(x + h) - f(x) \) is:

\[
\boxed{f(x + h) - f(x) = 4xh + 2h^2 - 5h}
\]

### Part (C) \( \frac{f(x + h) - f(x)}{h} \):

Finally, divide the result from part (B) by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 - 5h}{h}
\]

Simplify the expression by dividing each term by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = 4x + 2h - 5
\]

So, the simplified expression for \( \frac{f(x + h) - f(x)}{h} \) is:

\[
\boxed{\frac{f(x + h) - f(x)}{h} = 4x + 2h - 5}
\]

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Would you like further details or have any questions?

Here are some related questions:
1. How do you find the derivative of a function using the difference quotient?
2. What is the significance of the term "h" approaching zero in calculus?
3. How would you find the derivative of \( f(x) = 2x^2 - 5x + 8 \) using the power rule?
4. Can you explain why the function simplifies in each step?
5. What are some real-world applications of derivatives?

**Tip:** When simplifying expressions, always distribute and combine like terms carefully to avoid errors.

Learn how to simplify the function f(x) = 2x^2 - 5x + 8 for f(x + h) using the difference quotient method. This problem explores quadratic functions and algebraic manipulation techniques, suitable for high school students.

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