How many real solutions are there for $x$ in the following equation: $$(x - 5x + 12)^2 + 1 = -|x|$$
I notice that the equation has a squared term and an absolute value term, both of which are never negative. That means that the left-hand side of the equation is always positive, since it is the sum of a square and 1. The right-hand side of the equation is always non-positive, since it is the opposite of an absolute value. Therefore, the equation can never be true, because a positive number cannot equal a non-positive number. So there are no real solutions for $x$. 