show me how you simplify f(a+h) = 2(a+h)^2-5(a+h)+1

Let's simplify the expression step by step for \( f(a+h) = 2(a+h)^2 - 5(a+h) + 1 \).

### Step 1: Expand \( 2(a+h)^2 \)
The first part is \( (a+h)^2 \), so we expand this as:

\[
(a+h)^2 = a^2 + 2ah + h^2
\]

Now multiply this by 2:

\[
2(a^2 + 2ah + h^2) = 2a^2 + 4ah + 2h^2
\]

### Step 2: Expand \( -5(a+h) \)
Next, expand \( -5(a+h) \):

\[
-5(a+h) = -5a - 5h
\]

### Step 3: Put everything together
Now, combine all the terms from Step 1 and Step 2, and include the constant \( +1 \):

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

### Final Simplified Expression:

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

This is the simplified form of the given function.

Would you like further details or have any questions?

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Here are 5 related questions:
1. How would you find \( f(a) \) based on this function?
2. What happens when you plug in specific values for \( h \) into the simplified form?
3. How do you find the derivative of \( f(a+h) \) with respect to \( h \)?
4. Can you solve for the zeros of the function \( f(a+h) \)?
5. How does the simplification change if the leading coefficient was 3 instead of 2?

**Tip**: When expanding binomials like \( (a+h)^2 \), always apply the distributive property carefully to avoid mistakes in combining terms!

show me how you simplify f(a+h) = 2(a+h)^2 - 5(a+h) + 1

Learn how to simplify the polynomial expression f(a+h) = 2(a+h)^2 - 5(a+h) + 1 with a detailed step-by-step solution. This problem involves algebraic concepts such as binomial expansion and the distributive property. Suitable for students in Grades 9-11.

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