This looks like a geometric series, where each term is obtained by multiplying the previous term by $x$. I recall that the sum of a geometric series with first term $a$ and common ratio $r$ is given by $$S_n = \frac{a(1 - r^n)}{1 - r},$$ where $n$ is the number of terms. In this case, $a = x = -1$ and $r = x = -1$, and there are $n = 2011$ terms in the series.
[* so I can plug these values into the formula and simplify: $$S_{2011} = \frac{-1(1 - (-1)^{2011})}{1 - (-1)} = \frac{-1(1 + 1)}{2} = -1.$$ 
Let $x = -1$. Find $$x + x^2 + x^3 + \cdots + x^{2010} + x^{2011}.$$