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Let's solve the given problem step by step.

The function is given as:

\[
f(x) = x^2 + 6x
\]

### (a) Find \( f(x + h) \) and simplify.

We substitute \( x + h \) into \( f(x) \):

\[
f(x + h) = (x + h)^2 + 6(x + h)
\]

Now, expand and simplify:

\[
f(x + h) = (x^2 + 2xh + h^2) + 6(x + h)
\]
\[
f(x + h) = x^2 + 2xh + h^2 + 6x + 6h
\]

Thus, the simplified form of \( f(x + h) \) is:

\[
f(x + h) = x^2 + 2xh + h^2 + 6x + 6h
\]

### (b) Find \( \frac{f(x + h) - f(x)}{h} \) and simplify.

Now, let's find the difference between \( f(x + h) \) and \( f(x) \):

\[
f(x + h) - f(x) = (x^2 + 2xh + h^2 + 6x + 6h) - (x^2 + 6x)
\]

Simplify by canceling out terms:

\[
f(x + h) - f(x) = 2xh + h^2 + 6h
\]

Now, divide by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 + 6h}{h}
\]

Simplify:

\[
\frac{f(x + h) - f(x)}{h} = 2x + h + 6
\]

### Final Answer:
- (a) \( f(x + h) = x^2 + 2xh + h^2 + 6x + 6h \)
- (b) \( \frac{f(x + h) - f(x)}{h} = 2x + h + 6 \)

Would you like further details on any step, or have any questions?

Here are some related questions to expand your understanding:

1. What is the geometric interpretation of the difference quotient in part (b)?
2. How does this formula help in finding the derivative of \( f(x) \)?
3. Can you find the derivative of \( f(x) = x^2 + 6x \) using first principles?
4. What would \( f'(x) \) be if we applied basic derivative rules instead?
5. How does this process change if we have a more complex function, such as a cubic polynomial?

**Tip**: When solving for \( f(x+h) \), always simplify carefully by expanding all terms fully before performing cancellations.

Given f(x) = x^2 + 6x, (a) Find f(x+h) and simplify, (b) Find (f(x+h) - f(x))/h and simplify.

Step-by-step guide on finding and simplifying f(x+h) and the difference quotient for the function f(x) = x^2 + 6x. This exercise demonstrates critical algebraic techniques in manipulating polynomials and understanding the concept of rates of change.

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