Please do not think me petty; but this posting is unreadable.
Why can you simply type out the question in clear type?
There is no need to try to use symbols. sqrt{n^2+3n-6} is very clear as it stands.
Putting \$ around \sqrt{n^2+3n-6} gives $\sqrt{n^2+3n-6}$
OR at least take clear pictures that are orientated correctly.
If you follow those suggestions you are more likely to get help.
For example, you start with $\displaystyle N= \frac{1}{2}(n^2- n)$ and on the very next line have $\displaystyle N= 2(\frac{1}{2}(n^2- n)$ that has two errors! First there is an end parenthesis missing. Yes, I know that was just a "typo" but such things can make you work very hard to follow. Second, you have multiplied by 2 on the right but not on the left. Again, that is probably just an oversight since you have the "2N" in the next line but such errors can lead to greater mistakes later on.
Then you have $\displaystyle \sqrt{2N}= \sqrt{(n- n}$ where you have again dropped ")"- though strictly speaking you don't need the parentheses at all. More importantly the "$\displaystyle ^2$ is missing.
On the very next step, you still have the $\displaystyle \sqrt{}$ on the left but it has disappeared from the right! And you still have the "$\displaystyle n- n$" instead of "$\displaystyle n^2- n$".
It took me a while to figure out what in the world you had done to get the last line! You have apparently multiplied by n on the left side but added n on the right! And you are still missing the $\displaystyle n^2$. I agree with DenisB that you need to work with you teach on this.