Rút gọn biểu thức \(P = \left( {x - \frac{{{x^2} + {y^2}}}{{x + y}}} \right).\left( {\frac{{2x}}{y} + \frac{{4x}}{{x - y}}} \right):\frac{1}{y}\) (y ≠ 0, y ≠ x, y ≠ –x).

Với y ≠ 0, y ≠ x, y ≠ –x, ta có:

\(P = \left( {x - \frac{{{x^2} + {y^2}}}{{x + y}}} \right).\left( {\frac{{2x}}{y} + \frac{{4x}}{{x - y}}} \right):\frac{1}{y}\)

\( = \left[ {\frac{{x\left( {x + y} \right)}}{{x + y}} - \frac{{{x^2} + {y^2}}}{{x + y}}} \right].\left[ {\frac{{2x\left( {x - y} \right)}}{{y\left( {x - y} \right)}} + \frac{{4xy}}{{y\left( {x - y} \right)}}} \right]:\frac{1}{y}\)

\( = \left[ {\frac{{x\left( {x + y} \right) - {x^2} - {y^2}}}{{x + y}}} \right].\left[ {\frac{{2x\left( {x - y} \right) + 4xy}}{{y\left( {x - y} \right)}}} \right]:\frac{1}{y}\)

\( = \left[ {\frac{{{x^2} + xy - {x^2} - {y^2}}}{{x + y}}} \right].\left[ {\frac{{2{x^2} - 2xy + 4xy}}{{y\left( {x - y} \right)}}} \right]:\frac{1}{y}\)

\[ = \frac{{xy - {y^2}}}{{x + y}}.\frac{{2{x^2} + 2xy}}{{y\left( {x - y} \right)}}.y\]

\[ = \frac{{y\left( {x - y} \right)}}{{x + y}}.\frac{{2x\left( {x + y} \right)}}{{y\left( {x - y} \right)}}.y\] = 2xy.