https://file.aiquickdraw.com/mathbot/file/1727037465373pnmyyykt.jpg

The expression in the image is:

\[
\sum_{i=1}^{n}(i + 2)^3 - \sum_{i=3}^{n+1} i^3
\]

### Let's break this down:

1. The first summation, \(\sum_{i=1}^{n}(i + 2)^3\), expands by taking values of \(i\) from 1 to \(n\), and for each \(i\), we calculate \((i + 2)^3\).
   
2. The second summation, \(\sum_{i=3}^{n+1} i^3\), runs from \(i = 3\) to \(n + 1\), and we compute \(i^3\) for each value of \(i\).

### Next, simplify the expression:

- Notice that for the first summation, \((i + 2)^3\) is just the cube of \(i\) shifted by 2.
- We can adjust the limits and see how both summations overlap.

Now I will compute this expression more explicitly for general \(n\).

Let me calculate it for you.The given expression simplifies to:

\[
n^3 + 6n^2 + 12n + 8
\]

This is a cubic polynomial in terms of \(n\).

### Do you have any questions, or would you like further details?

Here are five related questions that might help you explore this further:

1. How do we generalize summations involving shifted terms like \((i + 2)^3\)?
2. What are the properties of the sum of cubes \(\sum_{i=1}^{n} i^3\)?
3. Can we derive a closed form for more complex summations?
4. What happens if we increase the exponent, say \((i + 2)^4\)?
5. How can we approximate such expressions for large \(n\)?

**Tip:** When simplifying expressions with summations, always look for overlaps in summation indices and limits to reduce terms efficiently.

The summation expression \(\sum_{i=1}^{n}(i + 2)^3 - \sum_{i=3}^{n+1} i^3\) was submitted.

Learn how to simplify the expression \(\sum_{i=1}^{n}(i + 2)^3 - \sum_{i=3}^{n+1} i^3\). This problem involves summations and algebraic manipulation of cubic expressions. Suitable for advanced high school or early university-level students.

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