jmc

Consider the quadratic formula $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Since the quadratic has exactly one root, then its discriminant must be 0. Thus, this gives us $$\begin{array}{rl}0& =b^2-4ac\\ \Rightarrow 0& =(c+1{)}^2-4c\\ \Rightarrow 0& =(c^2+2c+1)-4c\\ \Rightarrow 0& =c^2-2c+1\\ \Rightarrow 0& =(c-1{)}^2.\end{array}$$Since this expression is a perfect square, the only possible of value of $c$ is 1. Thus, the product of all possible values of $c$ is $\boxed{1}$.