\(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2=\left(x^2+4x+8+\dfrac{3}{2}x\right)^2-\dfrac{1}{4}x^2=\left(x^2+\dfrac{11}{2}x+8\right)^2-\left(\dfrac{1}{2}x\right)^2=\left(x^2+\dfrac{11}{2}x+8-\dfrac{1}{2}x\right)\left(x^2+\dfrac{11}{2}x+8+\dfrac{1}{2}x\right)=\left(x^2+5x+8\right)\left(x^2+6x+8\right)=\left(x+2\right)\left(x+4\right)\left(x^2+5x+8\right)\)

\(\left(x^2+4x+8\right)^2+3x\left(x^2+4x+8\right)+2x^2\)

\(=\left(x^2+4x+8\right)^2+x\left(x^2+4x+8\right)+2x\left(x^2+4x+8\right)+2x^2\)

\(=\left(x^2+4x+8\right)\left(x^2+5x+8\right)+2x\left(x^2+5x+8\right)\)

\(=\left(x^2+5x+8\right)\left(x+2\right)\left(x+4\right)\)

đặt x^2+x = y
=> y^2 - 2y - 15
= y^2 - 2y + 1 - 16

= ( y - 1 )^2 - 16

= ( y - 1 )^2 - 4^2

= ( y - 1 - 4 ) x ( y-1+4)

=(y -5) (y+3)

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

hi

`x^4+2x^3+5x^2+4x-12`

`=x^4-x^3+3x^3-3x^2+8x^2-8x+12x-12`

`=x^3(x-1)+3x^2(x-1)+8x(x-1)+12(x-1)`

`=(x-1)(x^3+3x^2+8x+12)`

`=(x-1)(x^3+2x^2+2x^2+4x+6x+12)`

`=(x-1)[x^2(x+2)+2x(x+2)+6(x+2)]`

`=(x-1)(x+2)(x^2+2x+6)`

\(x^4+2x^3+5x^2+4x-12\)

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

\(=\left(x-1\right)\left(x^3+3x^2+8x+12\right)\)

\(=\left(x-1\right)\left(x^2+2x^2+x^2+2x+6x+12\right)\)

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

Đặt x^2+2x=t =>3t^2-2t-1=3t^2-3t+t-1=3t(t-1)+(t-1)=(t-1)(3t+1)

=>(x^2+2x-1)(3x^2+6x+1)

Đặt x^2-3x-2=t =>(t+4)(t-4)+12=t-16+12=t-4=(t+2)(t-2)

=>(x^2-3x-2+2)(x^2-3x-2-2)=(x^2-3x)(x^2-3x-4)

(x + 1)(x + 2)(x + 3)(x + 4) - 24

= x⁴ + 10x³ + 35x² + 50x + 24 - 24

= x⁴ + 10x³ + 35x² + 50x

( x + 1 ). ( x + 2 ) ( x + 3 ) ( x + 4 ) - 24

= ( x² + 5x + 4 ) .( x² + 5x + 6 ) - 24

Đặt t = x² + 5x + 5 

=> ( t - 1 ). ( t + 1 ) - 24

= t² - 1 - 24 

= t² - 25

= ( t - 5 ). ( t + 5 )

= ( x² + 5x + 5 - 5 ) . ( x² + 5x + 5 + 5 )

= ( x² + 5x ) . ( x² + 5x + 10 )

= x. ( x + 5 ) . ( x² + 5x + 10 )

Đặt \(x^2+x+1=t\) 

Ta có: \(\left(x^2+x+1\right)^2+3x\left(x^2+x+1\right)+2x^2\)

\(=t^2+3xt+2x^2\)

\(=t^2+xt+2xt+2x\)

\(=t\left(t+x\right)+2x\left(t+x\right)\)

\(=\left(t+x\right)\left(t+2x\right)\)

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

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

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

Chúc bạn học tốt.

\(=\left(x^2+6x\right)\left(x^2+6x+8\right)-9\)

\(=\left(x^2+6x\right)^2+8\left(x^2+6x\right)-9\)

\(=\left(x^2+6x+9\right)\left(x^2+6x-1\right)\)

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