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The image you've uploaded asks to find the difference quotient of the function \( f(x) = x^2 + 5 \). The difference quotient formula is:

\[
\frac{f(x+h) - f(x)}{h}, \quad h \neq 0
\]

Let's go through the steps to find the difference quotient.

### Step 1: Compute \( f(x+h) \)
Given \( f(x) = x^2 + 5 \), we substitute \( x+h \) into the function:

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

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

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

Thus:

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

### Step 2: Compute \( f(x+h) - f(x) \)
Now, subtract \( f(x) = x^2 + 5 \) from \( f(x+h) \):

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

Simplifying:

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

### Step 3: Compute the difference quotient
Now, divide the result by \( h \):

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

Simplify the expression:

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

Thus, the difference quotient is:

\[
\boxed{2x + h}
\]

Would you like more details on any step? Here are some related questions you might find useful:

1. What is the difference quotient used for?
2. How do you find the derivative using the difference quotient?
3. How does the difference quotient relate to the slope of a secant line?
4. What happens to the difference quotient as \( h \) approaches zero?
5. Can you use the difference quotient for functions other than polynomials?

**Tip:** The difference quotient is a critical concept in calculus, as it leads directly to the definition of the derivative!

Find the difference quotient of f(x) = x^2 + 5.

Learn how to find the difference quotient of the function f(x) = x^2 + 5 with a detailed step-by-step solution. This problem covers key algebraic concepts and function manipulation, suitable for students preparing for calculus.

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