\[\frac{x^{4} + 2x^{6} + x^{7}}{2 + x + x^{- 2}} =\]
\[= \frac{x^{4}\left( 1 + 2x^{2} + x^{3} \right)}{2 + x + \frac{1}{x^{2}}} =\]
\[= \frac{x^{4}\left( 1 + 2x^{2} + x^{3} \right)}{\frac{2x^{2} + x^{3} + 1}{x^{2}}} =\]
\[= \frac{x^{4}\left( 1 + 2x^{2} + x^{3} \right) \cdot x²}{2x² + x³ + 1} = x^{6}\]