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!