Решите уравнение: (x+3)^4+(x+5)^4=272.

\[(x + 3)^{4} + (x + 5)^{4} = 272\]

\[t = \frac{x + 3 + x + 5}{2} = \frac{2x + 8}{2} =\]

\[= x + 4\]

\[(t - 1)^{4} + (t + 1)^{4} = 272\]

\[2t^{4} + 12t^{2} - 270 = 0\ \ \ \ \ \ |\ :2\]

\[t^{4} + 6t^{2} - 135 = 0\]

\[y = t^{2};\ \ \ \ \ \ y \geq 0.\]

\[y^{2} + 6y - 135 = 0\]

\[D = 6^{2} - 4 \cdot 1 \cdot ( - 135) =\]

\[= 36 + 540 = 576\]

\[y_{1} = \frac{- 6 + \sqrt{576}}{2} = \frac{- 6 + 24}{2} =\]

\[= \frac{18}{2} = 9\]

\[y_{2} = \frac{- 6 - \sqrt{576}}{2} = \frac{- 6 - 24}{2} =\]

\[= \frac{- 30}{2} = - 15\ (не\ подходит).\]

\[t^{2} = 9\]

\[Ответ:\ - 1;\ - 7.\]