$$\frac{{{a^2} - {b^2} - 2bc - {c^2}}}{{{a^2} + {b^2} + 2ab - {c^2}}}$$     is equivalent to = ?

$$\eqalign{ & \frac{{{a^2} - {b^2} - 2bc - {c^2}}}{{{a^2} + {b^2} + 2ab - {c^2}}}\, \cr & = \frac{{{a^2} - ({b^2} + 2bc + {c^2})}}{{({a^2} + {b^2} + 2ab) - {c^2}}} \cr & = \frac{{{a^2} - {{(b + c)}^2}}}{{{{(a + b)}^2} - {c^2}}} \cr & = \frac{{\left( {a + b + c} \right)\left( {a - b - c} \right)}}{{\left( {a + b + c} \right)\left( {a + b - c} \right)}} \cr & = \frac{{a - b - c}}{{a + b - c}} \cr} $$