You Do Not Need A Calculator

Relevant wiki: Multiplying and Dividing Monomials

Assume $x = 2016$ , so

$\begin{aligned} \dfrac{2017 \times (2016^2 -16) \times 2015}{2020 \times (2016^2 - 1)} &= \dfrac{(x+1) \times (x^2 - 16) \times (x-1)}{(x+4)\times(x^2 -1)}\\ & = \dfrac{(x^2 - 1) \times (x+4) \times (x-4)}{(x+4)\times(x^2 -1)}\\ & = x - 4\\ \end{aligned}$

So the answer is $2016 - 4 = \color{#D61F06}{\boxed{2012}}$

$\begin{aligned} Q & = \frac {(2017)\color{#3D99F6}{(2016^2-16)}(2015)}{(2020)\color{#D61F06}{(2016^2-1)}} \\ & = \frac {(2017)\color{#3D99F6}{(2016-4)(2016+4)}(2015)}{(2020)\color{#D61F06}{(2016-1)(2016+1)}} \\ & = \frac {(2017)\color{#3D99F6}{(2012)(2020)}(2015)}{(2020)\color{#D61F06}{(2015)(2017)}} \\ & = \frac {\cancel{(2017)}(2012)\cancel{(2020)}\cancel{(2015)}}{\cancel{(2020)}\cancel{(2015)}\cancel{(2017)}} \\ & = \boxed{2012} \end{aligned}$