<s>[INST] Sum of all numbers x such that $(3x^2+9x-2012)^{(x^3-2012x^2-10x+1)} = 1$? What is the sum of all $x$ such that $(3x^2 + 9x - 2012)^{(x^3-2012x^2-10x+1)} = 1$?
 [/INST] Note that $x^3-2012x^2-10x+1$ has no rational root (it would have to be $\pm1$, which can be checked explicitly).
Also note that yb polynomial division
$$\begin{align}x^3-2012x^2-10x+1 &= (3x^2+9x-2012)\cdot\frac{x-2015}3 + \frac{20117 x -4054177}3, \\
x^3-2012x^2-10x+k &= (3x^2+9x-2013)\cdot\frac{x-2015}3 + 6706 x -1352065+k, \\
x^3-2012x^2-10x+1 &= (3x^2+9x-2011)\cdot\frac{x-2015}3 + \frac{20116 x -4052162}3. \end{align}$$
The first implies that $3x^2+9x-2012$ and $x^3-2012x^2-10x+1$ cannot both be zero as that would lead to a rational root. 
Similarly, the third implies that we cannot have base $=1$ and exponent $=0$ at the same time.
Finally, the second equation shows that an $x$ for which the base is $-1$ and the exponent an integer $1-k$, again $x$ must be rational. However, $3x^2+9x-2013$ has no rational solutions.
Therefore, no weird special cases occur and the desired sum is simply the sum of the distinct(!) roots of $x^3-2012x^2-10x+1$ and the distinct(!) roots of $3x^2+9x-2013$. These can be read directly from the coefficients so that we obtain $$2012-\frac 93=2009.$$
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