b. \(a^3+6a^2+12a+8\\ =a^3+3.a^2.2+3.2^2.a+2^3\\ =\left(a+2\right)^3\)

a: \(=30a\left(a^2+4\right)-18b\left(a^2+4\right)\)

\(=6\left(a^2+4\right)\left(5a-3b\right)\)

b: \(=\left(a+2\right)^3\)

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. \(=\left(1-x\right)^3\)

\(1,\\ a,=4\left(x-2\right)^2+y\left(x-2\right)=\left(4x-8+y\right)\left(x-2\right)\\ b,=3a^2\left(x-y\right)+ab\left(x-y\right)=a\left(3a+b\right)\left(x-y\right)\\ 2,\\ a,=\left(x-y\right)\left[x\left(x-y\right)^2-y-y^2\right]\\ =\left(x-y\right)\left(x^3-2x^2y+xy^2-y-y^2\right)\\ b,=2ax^2\left(x+3\right)+6a\left(x+3\right)\\ =2a\left(x^2+3\right)\left(x+3\right)\\ 3,\\ a,=xy\left(x-y\right)-3\left(x-y\right)=\left(xy-3\right)\left(x-y\right)\\ b,Sửa:3ax^2+3bx^2+ax+bx+5a+5b\\ =3x^2\left(a+b\right)+x\left(a+b\right)+5\left(a+b\right)\\ =\left(3x^2+x+5\right)\left(a+b\right)\\ 4,\\ A=\left(b+3\right)\left(a-b\right)\\ A=\left(1997+3\right)\left(2003-1997\right)=2000\cdot6=12000\\ 5,\\ a,\Leftrightarrow\left(x-2017\right)\left(8x-2\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=2017\\x=\dfrac{1}{4}\end{matrix}\right.\\ b,\Leftrightarrow\left(x-1\right)\left(x^2-16\right)=0\Leftrightarrow\left[{}\begin{matrix}x=1\\x=4\\x=-4\end{matrix}\right.\)

\(-8x^3+1=1^3-\left(2x\right)^3=\left(1-2x\right)\left(1+2x+4x^2\right)\)

a) \(=x^4-14x^2+40-72=x^4-14x^2-32=\left(x-4\right)\left(x+4\right)\left(x^2+2\right)\)

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

c) \(=x^4+3x^3-3x^2+3x^3+9x^2-9x+x^2+3x-3-5=x^4+6x^3+7x^2-6x-8=\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x+4\right)\)

a: Ta có: \(\left(x^2-4\right)\left(x^2-10\right)-72\)

\(=x^4-14x^2-32\)

\(=\left(x^2-16\right)\left(x^2+2\right)\)

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

b: Ta có: \(\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+1\)

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

\(=\left(x^2+5x\right)^2+10\left(x^2+5x\right)+24+1\)

\(=\left(x^2+5x+1\right)^2\)

Bài 2:

1)  \(x^2-4x+4=\left(x-2\right)^2\)

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

3) \(1-8x^3=\left(1-2x\right)\left(1+2x+4x^2\right)\)

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

5) \(\dfrac{1}{25}x^2-64y^2=\left(\dfrac{1}{5}x-8y\right)\left(\dfrac{1}{5}x+8y\right)\)

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

Bài 2:

7) \(x^3+\dfrac{1}{27}=\left(x+\dfrac{1}{3}\right)\left(x^2+\dfrac{1}{3}x+\dfrac{1}{9}\right)\)

8) \(x^3+64=\left(x+4\right)\left(x^2+4x+16\right)\)

9) \(\left(a+b\right)^2-\left(2a-b\right)^2=\left(a+b+2a-b\right)\left(a+b-2a+b\right)=3a\left(-a+2b\right)\)

10) \(\left(a+b\right)^2-\left(a-b\right)^2=\left(a+b+a-b\right)\left(a+b-a+b\right)=2a\cdot2b=4ab\)

11) \(\left(a+b\right)^3+\left(a-b\right)^3=\left(a+b+a-b\right)\left[\left(a+b\right)^2+\left(a+b\right)\left(a-b\right)+\left(a-b\right)^2\right]\)

\(=2a\left(a^2+2ab+b^2+a^2-b^2+a^2-2ab+b^2\right)\)

\(=2a\left(3a^2+b^2\right)\)

12) \(\left(6x-1\right)^2-\left(3x+2\right)^2=\left(6x-1+3x+2\right)\left(6x-1-3x-2\right)=\left(9x+1\right)\left(3x-3\right)\)

1:

1: ,4x^2-6x=2x(2x-3)

2: 9x^3y^2+3x^2y^2=3x^2y^2(3x+1)

3: x^3+2x^2+3x=x(x^2+2x+3)

4: 2x^2-4x=2x(x-2)

5: 3x-6y=3(x-2y)

6: x^2-3x=x(x-3)

7: 6x^2y+4xy^2+2xy

=2xy(3x+2y+1)

8: 5x^2(x-2y)-15x(x-2y)

=(x-2y)(5x^2-15x)

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

9: =3(x-y)+5y(x-y)

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

10: =(x-1)(3x+5)

11: =2(2x-1)-3(2x-1)

=-(2x-1)

`a^2 + ab + 2a + 2b = a(a+2) + b(a+2) = (a+b)(a+2)`

7a/

$x^3y+x-y-1=(x^3y-y)+(x-1)=y(x^3-1)+(x-1)$

$=y(x-1)(x^2+x+1)+(x-1)=(x-1)[y(x^2+x+1)+1]$

$=(x-1)(x^2y+xy+y+1)$

7b/

$x^2(x-2)+4(2-x)=x^2(x-2)-4(x-2)=(x-2)(x^2-4)$

$=(x-2)(x-2)(x+2)=(x-2)^2(x+2)$
7c/

$x^3-x^2-20x=x(x^2-x-20)=x[(x^2+4x)-(5x+20)]$

$x[x(x+4)-5(x+4)]=x(x+4)(x-5)$

7d/

$(x^2+1)^2-(x+1)^2=[(x^2+1)-(x+1)][(x^2+1)+(x+1)]$

$=(x^2-x)(x^2+x+2)$

$=x(x-1)(x^2+x+2)$

7e/

$6x^2-7x+2=(6x^2-3x)-(4x-2)=3x(2x-1)-2(2x-1)=(2x-1)(3x-2)$

7f/

$x^4+8x^2+12=(x^4+6x^2)+(2x^2+12)=x^2(x^2+6)+2(x^2+6)$

$=(x^2+6)(x^2+2)$

7g/

$(x^3+x+1)(x^3+x)-2=(t+1)t-2$ (đặt $x^3+x=t$)

$=t^2+t-2=(t^2+2t)-(t+2)=t(t+2)-(t+2)$

$=(t+2)(t-1)=(x^3+x+2)(x^3+x-1)$

$=[(x^3+x^2)-(x^2+x)+(2x+2)](x^3+x-1)$

$=[x^2(x+1)-x(x+1)+2(x+1)](x^3+x-1)$

$=(x+1)(x^2-x+2)(x^3+x-1)$

7h.

$(x+1)(x+2)(x+3)(x+4)+1$

$=[(x+1)(x+4)][(x+2)(x+3)]+1=(x^2+5x+4)(x^2+5x+6)+1$

$=t(t+2)+1$ (đặt $x^2+5x+4=t$)

$=t^2+2t+1=(t+1)^2=(x^2+5x+5)^2$
7i/

Đặt $x^2+2=a$ thì:

$-(x^2+2)^2+4x(x^2+2)-3x^2=-a^2+4xa-3x^2$

$=-(a^2-ax)+(3ax-3x^2)$

$=-a(a-x)+3x(a-x)=(a-x)(3x-a)$

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

$=-(x^2-x+2)(x^2-3x+2)=-(x^2-x+2)(x-1)(x-2)$