Решите уравнение (x^2+5x+6)(x^2+5x+4)=840.

Ответ:

\[\left( x^{2} + 5x + 6 \right)\left( x^{2} + 5x + 4 \right) = 840\]

\[Пусть\ t = x^{2} + 5x + 4:\]

\[t(t + 2) = 840\]

\[t^{2} + 2t - 840 = 0\]

\[D = 1 + 840 = 841.\]

\[t_{1} = - 1 + 29 = 28;\ \ \ \]

\[t_{2} = - 1 - 29 = - 30.\]

\[Подставим:\]

\[1)\ x^{2} + 5x + 4 = 28\]

\[x^{2} + 5x - 24 = 0\]

\[x_{1} + x_{2} = - 5;\ \ \ x_{1} \cdot x_{2} = - 24\]

\[x_{1} = - 8;\ \ \ x_{2} = 3.\]

\[2)\ x^{2} + 5x + 4 = - 30\]

\[x^{2} + 5x + 34 = 0\]

\[D = 25 - 144 = - 119 < 0\]

\[корней\ нет.\]

\[Ответ:x = - 8;\ \ x = 3.\]