\(A=x^4-14x^3+71x^2-154x+120\)

\(=x^3\left(x-2\right)-12x^2\left(x-2\right)+47x\left(x-2\right)-60\left(x-2\right)\)

\(=\left(x-2\right)\left(x^3-12x^2+47x-60\right)\)

\(=\left(x-2\right)\left[x^2\left(x-3\right)-9x\left(x-3\right)+20\left(x-3\right)\right]\)

\(=\left(x-2\right)\left(x-3\right)\left(x^2-9x+20\right)=\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(x-5\right)\)

b, Vì A là tích của 4 số nguyên liên tiếp nên A chia hết cho 24

\(x^4-14x^3+71x^2-154x+120\)

\(=x^4-2x^3-12x^3+24x^2+47x^2-94x-60x+120\)

\(=x^3\left(x-2\right)-12x^2\left(x-2\right)+47x\left(x-2\right)-60\left(x-2\right)\)

\(=\left(x-2\right)\left(x^3-12x^2+47x-60\right)\)

\(=\left(x-2\right)\left(x^3-3x^2-9x^2+27x+20x-60\right)\)

\(=\left(x-2\right)\left[x^2\left(x-3\right)-9x\left(x-3\right)+20\left(x-3\right)\right]\)

\(=\left(x-2\right)\left(x-3\right)\left(x^2-9x+20\right)\)

\(=\left(x-2\right)\left(x-3\right)\left(x^2-4x-5x+20\right)\)

\(=\left(x-2\right)\left(x-3\right)\left[x\left(x-4\right)-5\left(x-4\right)\right]\)

\(=\left(x-2\right)\left(x-3\right)\left(x-4\right)\left(x-5\right)\)

$x^4 - 14x^3 + 71x^2 - 154x + 120$
$= (x^4 - 14x^3 + 71x^2 - 154x + 121) - 1$
$= (x^2 - 7x + 11)^2 - 1$
$= (x^2 - 7x + 10)(x^2 - 7x + 12)$
$= (x^2 - 5x - 2x + 10)(x^2 - 3x - 4x + 12)$
$= (x - 5)(x - 2)(x - 3)(x - 4)$

$=(x-2)(x-3)(x-4)(x-5)$

\(x^4-6x^3+12x^2-14x+3\) 

= \(x^4-4x^3+x^2-2x^3+8x^2-2x+3x^2-12x+3\) 

= \(x^2\left(x^2-4x+1\right)-2x\left(x^2-4x+1\right)+3\left(x^2-4x+1\right)\) 

= \(\left(x^2-4x+1\right)\left(x^2-2x+3\right)\)

\(=x^4-4x^3+x^2-2x^3+8x^2-2x+3x^2-12x+3\)

\(=x^2\left(x^2-4x-1\right)-2x\left(x^2-4x-1\right)+3\left(x^2-4x+1\right)\)

\(=\left(x^2-2x+3\right)\left(x^2-4x-1\right)\)

\(3x^3+5x^2-14x+4\\ =\left(3x^3-x^2\right)+\left(6x^2-2x\right)-\left(12x-4\right)\\ =x^2\left(3x-1\right)+2x\left(3x-1\right)-4\left(3x-1\right)\\ =\left(x^2+2-4\right)\left(3x-1\right)\)

\(b,=x^4-2x^3-x^3+2x^2+3x^2-6x-3x+6\\ =\left(x-2\right)\left(x^3-x^2+3x-3\right)\\ =\left(x-2\right)\left(x-1\right)\left(x^2+3\right)\\ c,=x^4-2x^3+4x^3-8x^2+4x^2-8x+3x-6\\ =\left(x-2\right)\left(x^3+4x^2+4x+3\right)\\ =\left(x-2\right)\left(x^3+3x^2+x^2+3x+x+3\right)\\ =\left(x-2\right)\left(x+3\right)\left(x^2+x+1\right)\)