Convert \(f\left( x \right) = x^2 - 2x - 3\) to vertex form by completing the square.

\[\begin{array}{l}
 f\left( x \right) = x^2 - 2x - 3 \\
 f\left( x \right) = \left( {x^2 - 2x} \right) - 3 \\
 f\left( x \right) = \left( {x^2 - 2x + \left( {\frac{{ - 2}}{2}} \right)^2 - \left( {\frac{{ - 2}}{2}} \right)^2 } \right) - 3 \\
 f\left( x \right) = \left( {x^2 - 2x + 1} \right) - 3 - 1 \\
 f\left( x \right) = \left( {x - 1} \right)^2 - 4 \\
 \end{array}\]