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 $$\begin{array}{rl}{2017}^2-2015\times 2019& =n^2-(n-2)(n+2)\\ & =n^2-(n^2+2n-2n-4)\\ & =n^2-(n^2-4)\\ & =n^2-n^2+4\\ & =4.\end{array}$$

Using just numbers

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

Then $$\begin{array}{rl}{2017}^2-2015\times 2019& ={2017}^2-(2017-2)(2017+2)\\ & ={2017}^2-({2017}^2+2017\times 2-2\times 2017-2^2)\\ & ={2017}^2-({2017}^2-4)\\ & ={2017}^2-{2017}^2+4\\ & =4.\end{array}$$