Let $p(x)$ be a polynomial with the maximum number of minus signs. $p(x)$ cannot have more than 1005 minus signs, otherwise $p(1)<0$ and $p(2) \geq 2^{2010}-2^{2009}-\ldots-2-1=$ 1, which implies, by the Intermediate Value Theorem, that $p$ must have a root greater than 1. Let $p(x)=\frac{x^{2011}+1}{x+1}=x^{2010}-x^{2009}+x^{2008}-\ldots-x+1 .-1$ is the only real root of $x^{2011}+1=0$ but $p(-1)=2011$; therefore $p$ has no real roots. Since $p$ has 1005 minus signs, it is the desired polynomial.