Решите уравнение, используя введение новой переменной: (x^2+x+6)(x^2+x-4)=144.

\[\left( x^{2} + x + 6 \right)\left( x^{2} + x - 4 \right) = 144\]

\[Пусть\ a = x^{2} + x + 6\]

\[a(a - 10) = 144\]

\[a^{2} - 10a - 144 = 0\]

\[D = 25 + 144 = 169\]

\[a_{1} = 5 + 13 = 18;\ \ \]

\[a_{2} = 5 - 13 = - 8.\]

\[Подставим:\]

\[1)\ x^{2} + x + 6 = 18\]

\[x^{2} + x - 12 = 0\]

\[x_{1} + x_{2} = - 1;\ \ \ x_{1} \cdot x_{2} = - 12\]

\[x_{1} = - 4;\ \ \ x_{2} = 3.\]

\[2)\ x^{2} + x + 6 = - 8\ \]

\[x^{2} + x + 14 = 0\]

\[D = 1 - 56 = - 55 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 4;\ \ x = 3.\]