a) ( x 2  – 4x + 1)( x 2  – 2x + 3).

b) ( x 2  + 5x – 1)( x 2  + x – 1).

\(x^3-5x^2-14x\)

\(=x^3+2x^2-7x^2-14x\)

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

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

\(=x\left(x+2\right)\left(x-7\right)\)

\(x^3-7x-6\)

\(=x^3+x^2-x^2-x-6x-6\)

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

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

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

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

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

\(x^3-19x-30\)

\(=x^3-5x^2+5x^2-25x+6x-30\)

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

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

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

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

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

c: \(x^4+x^3-4x^2+x+1\)

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

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

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

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

a) a⁴ + a² - 2

a⁴ + 2a² - a² - 2

a².( a² + 2 ) - ( a² + 2 )

( a² - 1 ).( a² + 2 )

( a + 1 ).( a - 1 ).( a² +2 )

b) x⁴ + 4x² - 5

x⁴ + 5x² - x² - 5

x².( x² + 5 ) - ( x² + 5 )

( x² - 1 ).( x² + 5 )

( x + 1 ).( x - 1 ).( x² + 5 )

c) x³ - 19x - 30

x³ + 2x² - 2x² + 4x - 15x - 30

x²( x + 2 ) - 2x.( x + 2 ) - 15.( x + 2 )

( x + 2 ).( x² - 2x - 15 )

d) x³ - 7x - 6

x³ - 3x² + 3x² - 9x + 2x - 6

x².( x - 3 ) + 3x.( x - 3 ) + 2.( x - 3 )

( x - 3 ).( x² + 3x +2 )

( x - 3 ).( x² + 2x + x + 2 )

( x - 3 ).( x.( x + 2 ) + ( x + 2 )

( x + 1 ).( x + 2 ).( x - 3 )

e) x³ - 5x² - 14x

x³ - 7x² + 2x² - 14x

x².( x - 7 ) + 2x.( x - 7 )

( x - 7 ).( x² + 2x )

x.( x + 2 ).( x - 7 )

x^3-19x-30 
=x^3-25x+6x-30 
=x(x^2-25)+6(x-5) 
=x(x+5)(x-5)+6(x-5) 
=(x-5)(x^2+5x+6) 
=(x-5)(x^2+2x+3x+6) 
=(x-5)[x(x+2)+3(x+2)] 
=(x-5)(x+2)(x+3)

\(x^3-19x-30=x^3+2x^2-2x^2-4x-15x-30\)

\(\Rightarrow x^2\left(x+2\right)-2x\left(x+2\right)-15\left(x+2\right)\)

\(\Rightarrow\left(x+2\right)\left(x^2-2x-15\right)\)

\(\Rightarrow\left(x+2\right)\left(x+3\right)\left(x-5\right)\)

x^3 + 3x^2 - 3x^2 - 9x - 10x - 30

= x^2 ( x+3) - 3x ( x +3) -10(x+3)

=(x^2 -3x - 10)(x+3) 

=(x^2 - 5x + 2x -10)(x+3)

 =[x(x-5)+2(x-5)](x+3)

=(x-5)(x+2)(x+3)

\(a,x^2-5=x^2-\left(\sqrt{5}\right)^2=\left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)\)

\(b,x^4+x^3+x+1=x^3.\left(x+1\right)+\left(x+1\right)\)

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

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

\(c,x^3-19x-30=x^3-25x+6x-30\)

\(=x.\left(x^2-25\right)+6.\left(x-5\right)\)

\(=x.\left(x-5\right)\left(x+5\right)+6.\left(x-5\right)\)

\(=\left(x-5\right).\left[x\left(x+5\right)+6\right]\)

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

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

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

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

Ta có: \(x^2-19x-30=\frac{4x^2-76x-120}{4}\)

                                          \(=\frac{1}{4}.\left[\left(4x^2-76x+361\right)-481\right]\)

                                          \(=\frac{1}{4}.\left[\left(2x-19\right)^2-481\right]\)

                                          \(=\frac{1}{4}.\left(2x-19-\sqrt{481}\right).\left(2x-19+\sqrt{481}\right)\)

Nghiệm xấu nên phân tích khó :) Sửa thành x³ - 19x - 30 cho dễ

x³ - 19x - 30

= x³ + 3x² - 3x² - 9x - 10x - 30

= ( x³ + 3x² ) - ( 3x² + 9x ) - ( 10x + 30 )

= x²( x + 3 ) - 3x( x + 3 ) - 10( x + 3 )

= ( x + 3 )( x² - 3x - 10 )

= ( x + 3 )( x² + 2x - 5x - 10 )

= ( x + 3 )[ x( x + 2 ) - 5( x + 2 ) ]

= ( x + 3 )( x + 2 )( x - 5 )

\(x^2-19x-30=\left(x-\frac{19+\sqrt{481}}{2}\right)\left(x-\frac{19-\sqrt{481}}{2}\right)\)

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

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

\(a,=-\left(x-1\right)^3\left[=\left(1-x\right)^3\right]\\ b,=\left(1-x\right)^3\)

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

b. \(=\left(1-x\right)^3\)