If $$m + \frac{1}{{m - 2}} = 4{\text{,}}$$    find the value of $${\left( {m - 2} \right)^2}{\text{ + }}\frac{1}{{{{\left( {m - 2} \right)}^2}}}$$     is?

$$\eqalign{ & m + \frac{1}{{m - 2}} = 4 \cr & \Rightarrow m - 2 + \frac{1}{{m - 2}} = 2 \cr & \left( {{\text{Squaring the both sides}}} \right) \cr} $$
$$ \Rightarrow {\left( {m - 2} \right)^2}{\text{ + }}\frac{1}{{{{\left( {m - 2} \right)}^2}}} + 2 \times $$      $$\left( {m - 2} \right) \times $$  $$\frac{1}{{\left( {m - 2} \right)}}$$   $$ = 4$$
$$ \Rightarrow {\left( {m - 2} \right)^2}{\text{ + }}\frac{1}{{{{\left( {m - 2} \right)}^2}}} = 2$$