\(x^4-14x^3+71x^2-154x+120\)
\(=x^4-2x^3-12x^3+24x^2+47x^2-94x-60x+120\)
\(=x^3\left(x-2\right)-12x^2\left(x-2\right)+47x\left(x-2\right)-60\left(x-2\right)\)
\(=\left(x-2\right)\left(x^3-12x^2+47x-60\right)\)
\(=\left(x-2\right)\left(x^3-3x^2-9x^2+27x+20x-60\right)\)
\(=\left(x-2\right)\left[x^2\left(x-3\right)-9x\left(x-3\right)+20\left(x-3\right)\right]\)
\(=\left(x-2\right)\left(x-3\right)\left(x^2-9x+20\right)\)
\(=\left(x-2\right)\left(x-3\right)\left(x^2-4x-5x+20\right)\)
\(=\left(x-2\right)\left(x-3\right)\left[x\left(x-4\right)-5\left(x-4\right)\right]\)
\(=\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(x-5\right)\)
$x^4 - 14x^3 + 71x^2 - 154x + 120$
$= (x^4 - 14x^3 + 71x^2 - 154x + 121) - 1$
$= (x^2 - 7x + 11)^2 - 1$
$= (x^2 - 7x + 10)(x^2 - 7x + 12)$
$= (x^2 - 5x - 2x + 10)(x^2 - 3x - 4x + 12)$
$= (x - 5)(x - 2)(x - 3)(x - 4)$
$=(x-2)(x-3)(x-4)(x-5)$