\(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)
\(=\left(x+1\right)\left(x+4\right)\left(x+2\right)\left(x+3\right)-24\)
\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)-24\) (1)
Đặt \(t=x^2+5x+4\)
(1) \(=t\left(t+2\right)-24\)
\(=t^2+2t-24\)
\(=t^2+2t+1-25\)
\(=\left(t+1\right)^2-5^2\)
\(=\left(t+1-5\right)\left(t+1+5\right)\)
\(=\left(t-4\right)\left(t+6\right)\)
\(\Rightarrow\left(1\right)=\left(x^2+5x+4-4\right)\left(x^2+5x+4+6\right)\)
\(=\left(x^2+5x\right)\left(x^2+5x+10\right)\)
\(=x\left(x+5\right)\left(x^2+5x+10\right)\)
\(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)
\(=x\left(x+2\right)\left(x+3\right)\left(x+4\right)+\left(x+2\right)\left(x+3\right)\left(x+4\right)-24\)
\(=x^2\left(x+3\right)\left(x+4\right)+2x\left(x+3\right)\left(x+4\right)+x\left(x+3\right)\left(x+4\right)+2\left(x+3\right)\left(x+4\right)-24\)
\(=x^2\left(x+3\right)\left(x+4\right)+3x\left(x+3\right)\left(x+4\right)+2\left(x+3\right)\left(x+4\right)-24\)
\(=x^3\left(x+4\right)+3x^2\left(x+4\right)+3x^2\left(x+4\right)+9x\left(x+4\right)+2x\left(x+4\right)+6\left(x+4\right)-24\)
\(=x^3\left(x+4\right)+6x^2\left(x+4\right)+11x\left(x+4\right)+6\left(x+4\right)-24\)
\(=x^4+4x^3+6x^3+24x^2+11x^2+44x+6x+24-24\)
\(=x^4+10x^3+35x^2+50x\)
\(=x\left(x+5\right)\left(x^2+5x+10\right)\)