Let's calculate the last expression step by step based on the image you provided.

You are given the function:

$f(x) = 4x + 3$

Step 1: Calculate $f(a)$

Substitute $a$ into the function $f(x)$:

$f(a) = 4a + 3$

Step 2: Calculate $f(a + h)$

Substitute $a + h$ into the function $f(x)$:

$f(a + h) = 4(a + h) + 3 = 4a + 4h + 3$

Step 3: Find $\frac{f(a+h) - f(a)}{h}$

Now, subtract $f(a)$ from $f(a + h)$ and simplify:

$f(a+h) - f(a) = (4a + 4h + 3) - (4a + 3) = 4h$

Then, divide by $h$:

$\frac{f(a+h) - f(a)}{h} = \frac{4h}{h} = 4$

Thus, the final value is:

$\boxed{4}$

The result shown as 1 in the image might be a placeholder, but the actual calculation yields 4.

Would you like further explanation or details on any step?

Relative Questions:

1. What is the geometric interpretation of $\frac{f(a+h) - f(a)}{h}$?
2. How would the result change if the function was quadratic, say $f(x) = x^2$?
3. Can this expression be used to find the derivative of linear functions?
4. What happens to this limit when $h \to 0$?
5. How does this formula connect to the definition of a derivative?

Tip: The expression $\frac{f(a+h) - f(a)}{h}$ is known as the difference quotient, which plays a crucial role in defining derivatives.