How many real solutions are there for $x$ in the following equation: $$(x - 5x + 12)^2 + 1 = -|x|$$
We can see that $(x - 5x + 12)^2$ must be nonnegative.  Thus $(x - 5x + 12)^2 + 1 > 0$.  But clearly, $-|x|$ is nonpositive. Thus there are $\boxed{0}$ solutions to the given equation.