How do people perform mental arithmetic for complicated expressions?

$$\begin{align}\\&\frac{10^2+11^2+12^2+13^2+14^2}{365}\\&=\frac{(12-2)^2+(12-1)^2+12^2+(12+1)^2+(12+2)^2}{365}\\&=\frac{5\times 12^2+10}{365}\\&=\frac{5(144+2)}{5\times 73}\\&=2\end{align}$$

If you know your squares out to $14$ (which students used to memorize) and do some simple three-digit arithmetic in your head, you can see that

$$100+121+144=365$$ and $$169+196=365$$

I think you can see clearly here that if you let $12$ be equal to $x$, the expression would just then be

$$\frac{(x-2)^2+(x-1)^2+x^2+(x+1)^2+(x+2)^2}{365}$$

Do remember that if you square a binomial $(a+b)$ you would get $a^2+2ab+b^2$; thus if you replace $a$ by $x$ and $b$ by either $\pm 1$ or $\pm 2$ the middle terms would just cancel out mainly $2ab$. So you would be left with

$$\frac{(x^2+4)+(x^2+1)+x^2+(x^2+1)+(x^2+4)}{365}$$

Which then further simplifies into

$$\frac{5x^2+10}{365}$$

$$\frac{720+10}{365}$$

$$=2$$