Simplify: $\;$ $\left(\dfrac{\dfrac{1}{a} + \dfrac{1}{b + c}}{\dfrac{1}{a} - \dfrac{1}{b + c}}\right) \left(1 + \dfrac{b^2 + c^2 - a^2}{2bc}\right) \left(a + b + c\right)^{-2}$
$\left(\dfrac{\dfrac{1}{a} + \dfrac{1}{b + c}}{\dfrac{1}{a} - \dfrac{1}{b + c}}\right) \left(1 + \dfrac{b^2 + c^2 - a^2}{2bc}\right) \left(a + b + c\right)^{-2}$
$= \left(\dfrac{a + b + c}{b + c - a}\right) \left(\dfrac{2bc + b^2 + c^2 - a^2}{2bc}\right) \left(\dfrac{1}{a + b + c}\right)^2$
$= \dfrac{\left(b + c\right)^2 - a^2}{2bc \left(b + c + a\right) \left(b + c - a\right)}$
$= \dfrac{\left(b + c\right)^2 - a^2}{2bc \left[\left(b + c\right)^2 - a^2\right]}$
$= \dfrac{1}{2bc}$