If a² + b² + 2b + 4a + 5 = 0, then the value of $$\frac{{a - b}}{{a + b}}\,{\text{is?}}$$

$$\eqalign{ & {a^2} + {b^2} + 2b + 4a + 5 = 0 \cr & \Rightarrow {a^2} + {b^2} + 2b + 4a + 4 + 1 = 0 \cr & \Rightarrow {a^2} + 4a + 4 + {b^2} + 2b + 1 = 0 \cr & \Rightarrow {\left( {a + 2} \right)^2} + {\left( {b + 1} \right)^2} = 0 \cr & a + 2 = 0{\text{ }} \Rightarrow {\text{ a}} = - 2 \cr & b + 1 = 0\,\,\,\, \Rightarrow \,\,\,\,b = - 1 \cr & \frac{{a - b}}{{a + b}} \Rightarrow \frac{{ - 2 + 1}}{{ - 2 - 1}} \cr & \Rightarrow \frac{{ - 1}}{{ - 3}} = \frac{1}{3} \cr} $$