Now let's solve part (b) of the problem, which asks for $f(x+h) - f(x)$.

Given: $f(x) = x^2 + 7x + 6$ and from part (a), we have: $f(x+h) = x^2 + 7x + 6 + h^2 + 2xh + 7h$

Step-by-Step Solution:

1. Subtract $f(x)$ from $f(x+h)$: $f(x+h) - f(x) = \left(x^2 + 7x + 6 + h^2 + 2xh + 7h\right) - \left(x^2 + 7x + 6\right)$

2. Simplify the expression: When we subtract $f(x) = x^2 + 7x + 6$ from $f(x+h)$, the $x^2 + 7x + 6$ terms cancel out: $f(x+h) - f(x) = h^2 + 2xh + 7h$

Thus, the simplified expression for $f(x+h) - f(x)$ is: $f(x+h) - f(x) = h^2 + 2xh + 7h$

Would you like help with the next part or have any questions?

Here are five related questions to explore:

1. How can you further simplify the expression $\frac{f(x+h) - f(x)}{h}$?
2. What does the expression $f(x+h) - f(x)$ represent in terms of the slope of the function?
3. How would the result change if the function were $f(x) = x^2 + 3x + 5$?
4. What happens to $f(x+h) - f(x)$ as $h$ approaches 0?
5. Can you calculate the derivative of $f(x) = x^2 + 7x + 6$ directly using the power rule?

Tip: When simplifying expressions involving differences, always look for terms that cancel out to make the simplification process easier.