smc

Let's write out the multiplication, so that it becomes easier to see. $$(1+2x-x^2)(1+2x-x^2)(1+2x-x^2)(1+2x-x^2)$$ We can now see that the only way to get an $x^7$ is by taking three $-x^2$ and one $2x$. There are $\left(\genfrac{}{}{0ex}{}{4}{1}\right)=4$ ways to pick which term the $2x$ comes from, and the coefficient of each one is $(-1{)}^3(2)=-2$. Therefore, the coefficient of $x^7$ is $(4)(-2)=-8,\boxed{-8}$.