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.