Show that the equation
$$\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0$$
can be expressed as a quadratic equation in \(\mathrm { e } ^ { x }\).
Hence solve the equation \(\ln \left( 1 + \mathrm { e } ^ { - x } \right) + 2 x = 0\), giving your answer correct to 3 decimal places.