To understand the steps in your equation, let’s break it down systematically. The expression seems to undergo several transformations, which I’ll explain below.

1. Starting Expression: $\frac{(n+1)^2 - 1}{2(n+1)} - \frac{n^2 - 1}{2n}$

2. Combine Fractions: To combine the fractions, find a common denominator, which is $2n(n+1)$: $\frac{(n+1)^2 - 1}{2(n+1)} \cdot \frac{n}{n} - \frac{n^2 - 1}{2n} \cdot \frac{(n+1)}{(n+1)}$ This gives: $\frac{n((n+1)^2 - 1) - (n^2 - 1)(n+1)}{2n(n+1)}$

3. Simplify the Numerator: Now simplify the numerator:

   • Expand $(n+1)^2 - 1 = n^2 + 2n + 1 - 1 = n^2 + 2n$.
   • So, $n((n+1)^2 - 1) = n(n^2 + 2n) = n^3 + 2n^2$.
   • For $-(n^2 - 1)(n+1)$:
     • Expand: $-(n^2 - 1)(n+1) = -n^3 - n^2 + n + 1$.

   Combining both parts, the numerator becomes: $n^3 + 2n^2 - n^3 - n^2 + n - 1 = n^2 + n - 1$

4. Final Expression: Thus, you get: $\frac{n^2 + n - 1}{2n(n+1)} = \frac{n^2 + n + 1}{2n(n+1)}$ This step suggests you equate it to $0$, leading to the expression on the right side.

5. Conclusion: The simplified form should allow you to analyze properties or set it equal to something else as necessary for solving $n$.

Summary of Steps:

1. Combine fractions with a common denominator.
2. Expand and simplify the numerator.
3. Cancel terms where possible.
4. Set the equation if needed or further analyze.

If you want details on any specific step or have further questions, let me know!

Additional Questions:

1. What is the purpose of setting the expression equal to $n^2 + n + 1$?
2. Would you like to see more examples of combining fractions?
3. How do these transformations help in solving for $n$?
4. Are there specific types of equations you're interested in practicing?
5. Would you like a detailed explanation of expanding binomials?

Tip:

Always look for common factors or terms to simplify expressions before performing operations like addition or subtraction.