`(x + 2)(x + 3)(x + 4)(x + 5) - 24 = 0`
`[(x + 2)(x + 5)] [(x + 3)(x + 4)] - 24 = 0`
`(x^2 + 7x + 10)(x^2 + 7x + 12) - 24 = 0`
`(x^2 + 7x + 11 - 1)(x^2 + 7x + 11 + 1) - 24 = 0`
`(x^2 + 7x + 11) - 1 - 24 = 0`
`(x^2 + 7x + 11) - 25 = 0`
`(x^2 + 7x + 11 - 5)(x^2 + 7x + 11 + 5) = 0`
`(x^2 + 7x + 6)(x^2 + 7x + 16) = 0`
`=> x^2 + 7x + 6 = 0` hoặc `x^2 + 7x + 16 = 0`
Ta có: `x^2 + 7x + 16 = x^2 + 7x + 49/4 + 15/4 = (x + 7/2)^2 + 15/4`
Vì \(\left(x+\dfrac{7}{2}\right)^2\ge0\forall x\) nên \(\left(x+\dfrac{7}{2}\right)^2+\dfrac{15}{4}>0\)
`=> x^2 + 7x + 6 = 0`
`<=> x^2 + x + 6x + 6 = 0`
`<=> x(x + 1) + 6(x + 1) = 0`
`<=> (x + 1)(x + 6) = 0`
`<=> x + 1 = 0` hoặc `x + 6 = 0`
`<=> x = -1` hoặc `x = -6`
\(\Leftrightarrow\left(x^2+7x+12\right)\left(x^2+7x+10\right)-24=0\)
\(\Leftrightarrow\left(x^2+7x\right)^2+22\left(x^2+7x\right)+96=0\)
\(\Leftrightarrow\left(x^2+7x+6\right)\left(x^2+7x+16\right)=0\)
=>(x+1)(x+6)=0
=>x=-1 hoặc x=-6
\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24=0\\ \Leftrightarrow\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24=0\\ \Leftrightarrow\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24=0\\ \Leftrightarrow\left(x^2+7x+10\right)\left[\left(x^2+7x+10\right)+2\right]-24=0\\ \Leftrightarrow\left(x^2+7x+10\right)^2+2\left(x^2+7x+10\right)-24=0\\ \Leftrightarrow\left(x^2+7x+10\right)^2-4\left(x^2+7x+10\right)+6\left(x^2+7x+10\right)-24=0\)
\(\Leftrightarrow\left(x^2+7x+10\right)\left(x^2+7x+10-4\right)+6\left(x^2+7x+10-4\right)=0\\ \Leftrightarrow\left(x^2+7x+10+6\right)\left(x^2+7x+6\right)=0\\ \Leftrightarrow\left(x^2+7x+16\right)\left(x^2+7x+6\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+7x+16=0\\x^2+7x+6=0\end{matrix}\right.\\ \Leftrightarrow\Leftrightarrow\left[{}\begin{matrix}x^2+2.\dfrac{7}{2}x+\dfrac{49}{4}+\dfrac{15}{4}=0\\x^2+6x+x+6=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}\left(x+\dfrac{7}{2}\right)^2+\dfrac{15}{4}=0\left(vô.lí\right)\\x\left(x+6\right)+\left(x+6\right)=0\end{matrix}\right.\\ \Leftrightarrow\left(x+1\right)\left(x+6\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=-1\\x=-6\end{matrix}\right.\)
