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!