\(x\left(x+1\right)\left(x+2\right)\left(x+3\right)+1=0\)
\(\Leftrightarrow x\left(x+3\right)\left(x+1\right)\left(x+2\right)+1=0\)
\(\Leftrightarrow\left(x^2+3x\right)\left(x^2+3x+2\right)+1=0\)
Đặt \(x^2+3x=a\)
\(\Leftrightarrow a\left(a+2\right)+1=0\)
\(\Leftrightarrow a^2+2a+1=0\)
\(\Leftrightarrow\left(a+1\right)^2=0\)
\(\Leftrightarrow a=-1\)
\(\Rightarrow x^2+3x=-1\)
\(\Leftrightarrow x^2+3x+1=0\)
\(\Delta=3^2-4=9-4=5>0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x_1=\dfrac{-3+\sqrt{5}}{2}\\x_2=\dfrac{-3-\sqrt{5}}{2}\end{matrix}\right.\)
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