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The question asks to simplify expressions involving the function \( f(x) = 3x^2 - x + 2 \). Let's solve the problem step by step.

### Given:
\[
f(x) = 3x^2 - x + 2
\]

#### (a) Find \( f(a+1) \):

To compute \( f(a+1) \), substitute \( x = a+1 \) into the function:

\[
f(a+1) = 3(a+1)^2 - (a+1) + 2
\]

Now expand the expression:

\[
f(a+1) = 3(a^2 + 2a + 1) - (a + 1) + 2
\]
\[
f(a+1) = 3a^2 + 6a + 3 - a - 1 + 2
\]
\[
f(a+1) = 3a^2 + 5a + 4
\]

#### (b) Find \( f(a) + f(1) \):

First, calculate \( f(a) \) and \( f(1) \) individually:

- \( f(a) = 3a^2 - a + 2 \)
- \( f(1) = 3(1)^2 - 1 + 2 = 3 - 1 + 2 = 4 \)

Now sum these:

\[
f(a) + f(1) = (3a^2 - a + 2) + 4 = 3a^2 - a + 6
\]

#### (c) Find \( f(1+h) - f(1) \):

First, calculate \( f(1+h) \) by substituting \( x = 1+h \) into the function:

\[
f(1+h) = 3(1+h)^2 - (1+h) + 2
\]
\[
f(1+h) = 3(1 + 2h + h^2) - (1 + h) + 2
\]
\[
f(1+h) = 3 + 6h + 3h^2 - 1 - h + 2
\]
\[
f(1+h) = 3h^2 + 5h + 4
\]

Now, subtract \( f(1) = 4 \):

\[
f(1+h) - f(1) = (3h^2 + 5h + 4) - 4 = 3h^2 + 5h
\]

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Do you need further details on any step, or have additional questions? Here are 5 related questions to consider:

1. How do you expand binomials like \( (a+1)^2 \)?
2. What are the steps for simplifying polynomial expressions?
3. Can you describe the process for calculating \( f(x) \) for specific values of \( x \)?
4. How is the difference \( f(1+h) - f(1) \) related to the concept of derivatives?
5. What happens when the function is evaluated at negative numbers?

**Tip**: When simplifying polynomial expressions, always carefully expand and combine like terms to avoid mistakes.

Let f(x) = 3x^2 - x + 2. Find and simplify the following: (a) f(a+1), (b) f(a) + f(1), (c) f(1+h) - f(1)

Learn how to simplify expressions involving the function f(x) = 3x^2 - x + 2. This detailed solution walks through finding f(a+1), f(a) + f(1), and f(1+h) - f(1), covering key algebraic techniques like polynomial expansion and substitution.

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