( x² - x + 4 )( x² - x + 5 ) - 6

Đặt t = x² - x + 4

Đa thức đã cho trở thành

t( t + 1 ) - 6

= t² + t - 6

= t² - 2t + 3t - 6

= t( t - 2 ) + 3( t - 2 )

= ( t - 2 )( t + 3 )

= ( x² - x + 4 - 2 )( x² - x + 4 + 3 )

= ( x² - x + 2 )( x² - x + 7 )

\(x\left(x+1\right)\left(x^2+x-5\right)-6\)

\(=\left(x^2+x\right)\left(x^2+x-5\right)-6\)

\(=\left(x^2+x^2\right)^2-5\left(x^2+x\right)-6\)

\(=\left(x^2+x\right)^2+\left(x^2+x\right)-6\left(x^2+x\right)-6\)

\(=\left(x^2+x\right)\left(x^2+x+1\right)-6\left(x^2+x+1\right)\)

\(=\left(x^2+x-6\right)\left(x^2+x+1\right)\)

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

\(\left(x-3\right)\left(x-10\right)\left(x-5\right)\left(x-6\right)-24x^2\)

\(=\left(x^2+30-13x\right)\left(x^2+30-11x\right)-24x^2\)

\(=\left(x^2+30x-12x-x\right)\left(x^2+30x-12x+x\right)-24x^2\)

\(=\left(x^2+30-12x\right)^2-x^2-24x^2\)

\(=\left(x^2-12x+30\right)^2-\left(5x\right)^2\)

\(=\left(x^2-12x+30+5x\right)\left(x^2-12x+30-5x\right)\)

\(=\left(x^2-7x+30\right)\left(x^2-17x+30\right)\)

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

\(4\left(x+5\right)\left(x+6\right)\left(x+10\right)\left(x+12\right)-3x^2\)

\(=4\left[\left(x+5\right)\left(x+12\right)\right]\left[\left(x+6\right)\left(x+10\right)\right]-3x^2\)

\(=4\left(x^2+17x+60\right)\left(x^2+16x+60\right)-3x^2\)

\(=\left(2x^2+34x+120\right)\left(2x^2+32x+60\right)-3x^2\)

\(=\left(2x^2+33x+120\right)^2-x^2-3x^2\)

\(=\left(2x^2+33x+120-2x\right)\left(2x^2+33x+120+2x\right)\)

\(=\left(2x+15\right)\left(x+8\right)\left(2x^2+35x+120\right)\)

Lời giải:
$x-5\sqrt{x}+6=x-2\sqrt{x}-3\sqrt{x}+6$

$=\sqrt{x}(\sqrt{x}-2)-3(\sqrt{x}-2)$

$=(\sqrt{x}-2)(\sqrt{x}-3)$

\(x-5\sqrt{x}+6=\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)\)

\(4\left(x+5\right)\left(x+12\right)\left(x+6\right)\left(x+10\right)-3x^2\)

\(=2\left(x^2+60+17x\right).2\left(x^2+60+16x\right)-3x^2\)

\(=\left(2x^2+120+33x+x\right)\left(2x^2+120+33x-x\right)-3x^2\)

\(=\left(2x^2+120+33x\right)^2-x^2-3x^2\)

\(=\left(2x^2+120+33x\right)^2-4x^2\)

\(=\left(2x^2+120+33x+2x\right)\left(2x^2+120+33x-2x\right)\)

\(=\left(2x^2+35x+120\right)\left(2x^2+31x+120\right)\)

\(=\left(2x^2+35x+120\right)\left(x+8\right)\left(2x+15\right)\)