jmc

We group the addends inside the parentheses two at a time: $$\begin{array}{rl}-1+2-3+4-5+6-7+\dots +1000& =(-1+2)+(-3+4)+(-5+6)+\dots +(-999+1000)\\ & =\underset{\text{500 1’s}}{\underbrace{1+1+1+\dots +1}}\\ & =500.\end{array}$$ Then the desired answer is $4\times 500=\boxed{2000}$.