Bài `1`

\(a,5x^2-10xy=5x\left(x-2y\right)\\ b,3x\left(x-y\right)-6\left(x-y\right)=\left(x-y\right)\left(3x-6\right)\\ =3\left(x-y\right)\left(x-2\right)\\ c,2x\left(x-y\right)-4y\left(y-x\right)=2x\left(x-y\right)+4y\left(x-y\right)\\ =\left(x-y\right)\left(2x+4y\right)=2\left(x-y\right)\left(x+2y\right)\\ d,9x^2-9y^2=\left(3x\right)^2-\left(3y\right)^2=\left(3x-3y\right)\left(3x+3y\right)\\ f,xy-xz-y+z=\left(xy-xz\right)-\left(y-z\right)\\ =x\left(y-z\right)-\left(y-z\right)=\left(y-z\right)\left(x-1\right)\)

Bài `3`

\(a,3x^2+8x=0\\ \Leftrightarrow x\left(3x+8\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\3x+8=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\3x=-8\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x=-\dfrac{8}{3}\end{matrix}\right.\)

\(b,9x^2-25=0\\ \Leftrightarrow\left(3x\right)^2-5^2=0\\ \Leftrightarrow\left(3x-5\right)\left(3x+5\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}3x-5=0\\3x+5=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}3x=5\\3x=-5\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5}{3}\\x=-\dfrac{5}{3}\end{matrix}\right.\)

\(c,x^3-16x=0\\ \Leftrightarrow x\left(x^2-16\right)=0\\ \Leftrightarrow x\left(x-4\right)\left(x+4\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x-4=0\\x+4=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=0\\x=4\\x=-4\end{matrix}\right.\)

\(d,x^3+x=0\\ \Leftrightarrow x\left(x^2+1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x^2+1\in\varnothing\\x=0\end{matrix}\right.\Rightarrow x=0\)

a) \(2x^2+5x+2\)

\(=2x^2+4x+x+2\)

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

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

b) \(4x^2-4x-9y^2+12y-3\)

\(=\left(4x^2-4x+1\right)-\left(9y^2-12y+4\right)\)

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

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

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

c) \(x^4-2x^3-4x^2+4x-3\)

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

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

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

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

d) \(x^3-x+3x^2y+3xy^2+y^3-y\)

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

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

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

\(=\left(x+y\right)\left(x+y-1\right)\left(x+y+1\right)\)

\(x^2y-16y=y\left(x^2-16\right)=y\left(x-4\right)\left(x+4\right)\)

\(x^2-9+5xy-15y=\left(x-3\right)\left(x+3\right)+5y\left(x-3\right)=\left(x-3\right)\left(x+3+5y\right)\)

Lời giải:
a. $5x^2-10xy=5x(x-2y)$

b. $3x(x-y)-6(x-y)=(x-y)(3x-6)=3(x-y)(x-2)$
c. $2x(x-y)-4y(y-x)=2x(x-y)+4y(x-y)=(x-y)(2x+4y)=2(x-y)(x+2y)$

d. $9x^2-9y^2=9(x^2-y^2)=9(x-y)(x+y)$

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

f. $xy-xz-y+z=(xy-y)-(xz-z)=y(x-1)-z(x-1)=(x-1)(y-z)$

\(x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+3xyz\)

\(=\left(x^2y+xy^2+xyz\right)+\left(y^2z+yz^2+xyz\right)+\left(x^2z+xz^2+xyz\right)\)

\(=xy\left(x+y+z\right)+yz\left(y+z+x\right)+xz\left(x+z+y\right)\)

\(=\left(x+y+z\right)\left(xy+yz+xz\right)\)

Ta có:  9 x 2 - 1 = ( 3 x ) 2 - 1 2 = ( 3 x - 1 ) ( 3 x + 1 )

Ta có:  9 x 2 - 1 = ( 3 x ) 2 - 1 2 = ( 3 x - 1 ) ( 3 x + 1 )

a) 20x²y + 25xy2
= 5xy( 4x + 5y )
b) x² - 5x - 15y - 9y²
= x(x - 5) - 3y(5 + 3y)

(x + y)2 – 9x2 = (x + y)2 – (3x)2

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

= (4x + y)(-2x + y)

a) \(9x^2-16\)

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

\(=\left(3x-4\right)\left(3x+4\right)\)

b) \(x^2+4xy+4y^2-3x-6y\)

\(=\left(x^2+4xy+4y^2\right)-\left(3x+6y\right)\)

\(=\left[x^2+2\cdot x\cdot2y+\left(2y\right)^2\right]-3\left(x+2y\right)\)

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

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

#\(Toru\)