Решите уравнение: (x+1)(x+2)(x+3)(x+4)=360.

Ответ:

\[(x + 1)(x + 2)(x + 3)(x + 4) =\]

\[= 360\]

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

\[= 360\]

\[Пусть\ \left( x^{2} + 5x + 4 \right) = a:\]

\[a(a + 2) = 360\]

\[a^{2} + 2a - 360 = 0\]

\[D = 1 + 360 = 361\]

\[a_{1} = - 1 + 19 = 18;\ \ \ \]

\[a_{2} = - 1 - 19 = - 20\]

\[Подставим:\]

\[1)\ x^{2} + 5x + 4 = 18\]

\[x^{2} + 5x - 14 = 0\]

\[x_{1} + x_{2} = - 5;\ \ \ x_{1} \cdot x_{2} = - 14\]

\[x_{1} = - 7;\ \ \ x_{2} = 2.\]

\[2)\ x^{2} + 5x + 4 = - 20\]

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

\[D = 25 - 96 < 0\]

\[корней\ нет.\]

\[Ответ:x = - 7;\ \ x = 2.\]