Let's solve the given problem step by step.

Function Definition:

We are given $f(x) = 7x^2 - 2$.

Part a: Find $f(x + h)$:

Substitute $x + h$ into $f(x)$:

$f(x + h) = 7(x + h)^2 - 2$

Expand $(x + h)^2$:

$(x + h)^2 = x^2 + 2xh + h^2$

Substitute this into $f(x + h)$:

$f(x + h) = 7(x^2 + 2xh + h^2) - 2$

Distribute $7$:

$f(x + h) = 7x^2 + 14xh + 7h^2 - 2$

So, the simplified answer is:

$f(x + h) = 7x^2 + 14xh + 7h^2 - 2$

Part b: Find the difference quotient of $f$ and simplify:

The difference quotient is:

$\frac{f(x + h) - f(x)}{h}$

Substitute $f(x + h) = 7x^2 + 14xh + 7h^2 - 2$ and $f(x) = 7x^2 - 2$:

$f(x + h) - f(x) = (7x^2 + 14xh + 7h^2 - 2) - (7x^2 - 2)$

Simplify:

$f(x + h) - f(x) = 7x^2 + 14xh + 7h^2 - 2 - 7x^2 + 2$

$f(x + h) - f(x) = 14xh + 7h^2$

Divide by $h$:

$\frac{f(x + h) - f(x)}{h} = \frac{14xh + 7h^2}{h}$

Simplify by canceling $h$ (assuming $h \neq 0$):

$\frac{f(x + h) - f(x)}{h} = 14x + 7h$

Final Answers:

1. $f(x + h) = 7x^2 + 14xh + 7h^2 - 2$
2. Difference Quotient: $\frac{f(x + h) - f(x)}{h} = 14x + 7h$

Do you want further clarification or details? Here are some related questions:

1. What happens to the difference quotient as $h \to 0$?
2. Why is the difference quotient important in calculus?
3. Can you derive the derivative of $f(x) = 7x^2 - 2$ using these results?
4. What does the term $14x$ in the difference quotient represent geometrically?
5. How does expanding $(x + h)^2$ correctly simplify future calculations?

Tip: Always simplify expressions carefully to avoid mistakes in problems involving substitution or expansion!