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?

First of all rewrite your equation as $$|(x-6)^2-2|=c$$
Then take away the modulus, which means that you have to solve $$(x-6)^2-2-c=0\lor(x-6)^2-2+c=0$$
Since they are both quadratics they'll have 2 solutions each, for a total of 4 solutions, unless you make one of them a perfect square, by picking either $c=2$ or $c=-2$.
You should immediately see that only one of these 2 values makes sense
