Let's break down and solve the problem step by step, based on the image:

Given: $f(x) = 5x^2 + 3$

1. Finding $f(a)$:

To find $f(a)$, substitute $x = a$ into the function: $f(a) = 5a^2 + 3$

2. Finding $f(a + h)$:

Next, substitute $x = a + h$ into the function: $f(a + h) = 5(a + h)^2 + 3$ Expand the square: $f(a + h) = 5(a^2 + 2ah + h^2) + 3 = 5a^2 + 10ah + 5h^2 + 3$

3. Finding the difference quotient:

The difference quotient is given by: $\frac{f(a + h) - f(a)}{h}$ Substitute the expressions for $f(a + h)$ and $f(a)$: $\frac{(5a^2 + 10ah + 5h^2 + 3) - (5a^2 + 3)}{h}$ Simplify the numerator: $\frac{5a^2 + 10ah + 5h^2 + 3 - 5a^2 - 3}{h} = \frac{10ah + 5h^2}{h}$ Factor out $h$ from the numerator: $= \frac{h(10a + 5h)}{h} = 10a + 5h$

Final answers:

• $f(a) = 5a^2 + 3$
• $f(a + h) = 5a^2 + 10ah + 5h^2 + 3$
• $\frac{f(a + h) - f(a)}{h} = 10a + 5h$

Would you like further details on any part of this process?

Here are five related questions you can explore:

1. What is the significance of the difference quotient in calculus?
2. How do we use the difference quotient to find the derivative of a function?
3. How does expanding binomials relate to solving this problem?
4. What happens if we let $h \to 0$ in the difference quotient?
5. How would this problem change if the function were $f(x) = x^3 + 2$?

Tip: The difference quotient is the foundation for understanding derivatives, which measure the rate of change of a function.