Answer
\[ = {x^2} + 2xa + {a^2} - 2\]
Work Step by Step
\[\begin{gathered}
f\,\left( x \right) = {x^3} - 2x \hfill \\
\hfill \\
\frac{{f\,\left( x \right) - f\,\left( a \right)}}{{x - a}} = \frac{{{x^3} - 2x - {a^3} + 2a}}{{x - a}} \hfill \\
\hfill \\
factor \hfill \\
\hfill \\
= \frac{{{x^3} - {a^3} - 2\,\left( {x - a} \right)}}{{x - a}} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{{\,\left( {x - a} \right)\,\left( {{x^2} + 2xa + {a^2}} \right) - 2\,\left( {x - a} \right)}}{{x - a}} \hfill \\
\hfill \\
= \frac{{\,\left( {x - a} \right)\,\left( {{x^2} + 2xa + {a^2} - 2} \right)}}{{x - a}} \hfill \\
\hfill \\
= {x^2} + 2xa + {a^2} - 2 \hfill \\
\end{gathered} \]
