https://file.aiquickdraw.com/mathbot/file/1726034739040144xdjhi.jpg

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.

Find f(a), f(a + h), and the difference quotient (f(a + h) - f(a)) / h for the function f(x) = 5x^2 + 3.

This guide explains how to find f(a), f(a + h), and the difference quotient for the quadratic function f(x) = 5x^2 + 3. The solution covers algebraic expansion and simplification, focusing on the core concepts of quadratic functions and difference quotients.

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