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.