https://file.aiquickdraw.com/mathbot/file/1724965815528ocr5u6mc.jpg

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.

Explore how to solve the expression f(x+h) - f(x) for the quadratic function f(x) = x^2 + 7x + 6. Learn step-by-step with a detailed explanation, focusing on the difference quotient and its implications. Suitable for advanced high school students.

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