\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left(x^2+3x+2x+6\right)\left(x^2+5x+4x+20\right)-24\)
\(=\left(x^2+5x+6\right)\left(x^2+9x+20\right)-24\)
\(=\left(x^4+9x^3+20x^2+5x^3+45x^2+100x+6x^2+54x+120\right)-24\)
\(=x^4+14x^3+71x^2+154x+120-24\)
\(=x^4+14x^3+71x^2+154x+96\)
\(=x^4+13x^3+58x^2+96x+x^3+13x^2+58x+96\)
\(=x\left(x^3+13x^2+58x+96\right)+\left(x^3+13x^2+58x+96\right)\)
\(=\left(x^3+13x^2+58x+96\right)\left(x+1\right)\)
\(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)
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