Clever calculation

Answer: $4$

Using a symbol

Suppose we use a symbol to represent $2017$, such as $n$.

Then if $n=2017$, $2015=n-2$ and $2019=n+2$.

So $$\begin{split}2017^2-2015\times2019&=n^2-(n-2)(n+2)\\

&=n^2-(n^2+2n-2n-4)\\

&=n^2-(n^2-4)\\

&=n^2-n^2+4\\

&=4.\end{split}$$

Using just numbers

Write all the numbers in terms of $2017$, so $2015=2017-2$ and $2019=2017+2$.

Then $$\begin{split}2017^2-2015\times2019&=2017^2-(2017-2)(2017+2)\\

&=2017^2-(2017^2+2017\times2-2\times2017-2^2)\\

&=2017^2-(2017^2-4)\\

&=2017^2-2017^2+4\\

&=4.\end{split}$$