Решите уравнение (x^2+3x+4)(x^2+3x+9)=266.

Ответ:

\[\left( x^{2} + 3x + 4 \right)\left( x^{2} + 3x + 9 \right) = 266\]

\[Пусть\ t = x^{2} + 3x + 4:\]

\[t(t + 5) = 266\]

\[t^{2} + 5t - 266 = 0\]

\[D = 25 + 1064 = 1089 = 33^{2}\]

\[t_{1} = \frac{- 5 + 33}{2} = 14;\ \ \ \]

\[t_{2} = \frac{- 5 - 33}{2} = - 19.\]

\[Подставим:\]

\[1)\ x^{2} + 3x + 4 = 14\]

\[x^{2} + 3x - 10 = 0\]

\[x_{1} + x_{2} = - 3;\ \ \ \ x_{1} \cdot x_{2} = 10\]

\[x_{1} = - 5;\ \ \ x_{2} = 2.\]

\[2)\ x^{2} + 3x + 4 = - 19\]

\[x^{2} + 3x + 23 = 0\]

\[D = 9 - 92 = - 83 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 5;\ \ x = 2.\]