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 $(3x^2 + 9x - 2012)^{(x^3-2012x^2-10x+1)} = 1$?

Recall that $a^0 = 1$ for all real $a \neq 0$, and $1^b = 1$ for all real $b$.
$$(3x^2 + 9x - 2012)^{\large(x^3-2012x^2-10x+1)} = 1 \iff$$
$$ $$
$$x^3 \color{blue}{\bf - 2012}x^2 - 10 x + 1 = 0\tag{1}$$
or $$3x^2 + 9x - 2012 = 1 \iff 3x^2 + 9x - 2013 = 0 \iff \color{red}{\bf 1}\cdot x^2 + \color{red}{\bf 3}x - 671 = 0\quad\quad\quad\quad\tag{2}$$
There exist $3$ roots $x_1, x_2, x_3$ to $(1)$, and $2$ roots $x_4, x_5$ to (2).
The sum $S$ you want is $$S = \color{blue}{\bf x_1 + x_2 + x_3} + \color{red}{\bf x_4 + x_5}$$
To obtain this, note the colored coefficients: $${\bf Sum} = \color{blue}{\bf 2012} - \color{red}{\bf \frac 31} = {\bf 2009}.$$
