\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)=24\)
\(\Leftrightarrow\left(x+2\right)\left(x+5\right)\left(x+3\right)\left(x+4\right)-24=0\)
\(\Leftrightarrow\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24=0\)
\(\Leftrightarrow\left(x^2+5x+2x+10\right)\left(x^2+4x+3x+12\right)-24=0\)
\(\Leftrightarrow\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24=0\)
\(\Leftrightarrow\left(x^2+7x+10\right)\left[\left(x^2+7x+10\right)+2\right]-24=0\)
\(\Leftrightarrow\left(x^2+7x+10\right)^2+2\left(x^2+7x+10\right)-24=0\)
\(\Leftrightarrow\left(x^2+7x+10\right)^2-4\left(x^2+7x+10\right)+6\left(x^2+7x+10\right)-24=0\)
\(\Leftrightarrow\left[\left(x^2+7x+10\right)^2-4\left(x^2+7x+10\right)\right]+\left[6\left(x^2+7x+10\right)-24\right]=0\)
\(\Leftrightarrow\left(x^2+7x+10\right)\left[\left(x^2+7x+10\right)-4\right]+6\left[\left(x^2+7x+10\right)-4\right]=0\)
\(\Leftrightarrow\left[\left(x^2+7x+10\right)-4\right]\left[\left(x^2+7x+10\right)+6\right]=0\)
\(\Leftrightarrow\left(x^2+7x+6\right)\left(x^2+7x+16\right)=0\)
\(\Leftrightarrow\left(x^2+x+6x+6\right)\left(x^2+7x+16\right)=0\)
\(\Leftrightarrow\left[\left(x^2+x\right)+\left(6x+6\right)\right]\left(x^2+7x+16\right)=0\)
\(\Leftrightarrow\left[x\left(x+1\right)+6\left(x+1\right)\right]\left(x^2+7x+16\right)=0\)
\(\Leftrightarrow\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+1=0\\x+6=0\\x^2+7x+16=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=-6\\\left(x+\dfrac{7}{2}\right)^2+\dfrac{15}{4}>0\left(loai\right)\end{matrix}\right.\)
Vậy \(x=-1\) hoặc \(x=-6\)
