Đặt D=\(\left(x+1\right).\left(x+2\right).\left(x+3\right).\left(x+4\right)-24\)
\(=\left[\left(x+1\right).\left(x+4\right)\right].\left[\left(x+2\right).\left(x+3\right)\right]-24\)
\(=\left(x^2+4x+x+4\right).\left(x^2+3x+2x+6\right)-24\)
\(=\left(x^2+5x+4\right).\left(x^2+5x+6\right)-24\)
Đặt \(x^2+5x+4=t\Rightarrow x^2+5x+6=t+2\)
\(\Rightarrow D=t\left(t+2\right)-24=t^2+2t-24\)
\(D=t^2+6t-4t-24=\left(t^2+6t\right)-\left(4t+24\right)\)
\(D=t.\left(t+6\right)-4\left(t+6\right)=\left(t+6\right).\left(t-4\right)\)
Vì \(t=x^2+5x+4\) nên:
\(D=\left(x^2+5x+4+6\right).\left(x^2+5x+4-4\right)\)
\(D=\left(x^2+5x+10\right).\left(x^2+5x\right)\)
\(D=\left(x^2+5x+10\right).x.\left(x+5\right)\)
Vậy \(\left(x+1\right).\left(x+2\right).\left(x+3\right).\left(x+4\right)-24=\left(x^2+5x+10\right).x.\left(x+5\right)\)
Chúc bạn học tốt!!!
