For what real number $c$, this equation has exactly three solutions? For what real number c does the equation $|x^2 + 12x + 34| = c$ has exactly three solutions?

You need no fancy rewriting; to begin with, the equality
$$
|A|=B
$$
is equivalent to a set of two equalities
$$
A=B\quad\text{or}\quad A=-B
$$
so you get the solutions of your equation by putting together the solutions of
$$
x^2+12x+34=c
$$
and
$$
x^2+12x+34=-c
$$


*

*Note that no solution of the first can be a solution of the second, because this would happen only for $c=0$ and…

*Either equation can have zero, one or two solutions; what's the way to decide what case we're in?

*You need that the first equation has two solutions and the second equation has one solution or conversely
