a 3  –  a 2 x – ay + xy = ( a 3  –  a 2 x) – (ay – xy)

=  a 2 (a – x) – y(a – x) = (a – x)( a 2  – y)

b: \(x^2-2xy+y^2-z^2\)

\(=\left(x-y\right)^2-z^2\)

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

d: \(x^2+4x+3=\left(x+3\right)\left(x+1\right)\)

\(1,\\ a,=\left(x-y\right)^2-z^2=\left(x-y-z\right)\left(x-y+z\right)\\ b,=a^2\left(a-x\right)-y\left(a-x\right)=\left(a^2-y\right)\left(a-x\right)\\ c,=\left(x-y\right)^2-z\left(x-y\right)=\left(x-y\right)\left(x-y-z\right)\\ d,=x\left(x-2y\right)+t\left(x-2y\right)=\left(x+t\right)\left(x-2y\right)\\ 2,\\ \Rightarrow x^2-4x+4-x^2+9=6\\ \Rightarrow-4x=-7\Rightarrow x=\dfrac{7}{4}\\ 3,\\ a,x^2+2x+2=\left(x+1\right)^2+1\ge1>0\\ b,-x^2+4x-5=-\left(x-2\right)^2-1\le-1< 0\)

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

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

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

\(y^2-6y+9-z^2\)

\(=\left(y-3\right)^2-z^2\)

\(=\left(y-3-z\right)\left(y-3+z\right)\)

Bài 1:

c. \(a^3-a^2x-ay+xy\)

\(=a^2\left(a-x\right)-y\left(a-x\right)\)

\(=\left(a-x\right)\left(a^2-y\right)\)

Bài 2:

a. \(x^2-6x=0\)

\(x\left(x-6\right)=0\)

\(\Rightarrow\)\(\orbr{\begin{cases}x=0\\x=6\end{cases}}\)

b) \(x^3-2x2+x=0\)

\(x\left(x^2-3\right)=0\)

\(\Rightarrow\)\(\hept{\begin{cases}x=0\\x^2=3\end{cases}}\)

\(\Rightarrow\)\(\hept{\begin{cases}x=0\\x=\sqrt{3}\\x=-\sqrt{3}\end{cases}}\)

\(a^3\left(b-c\right)+b^3\left(c-a\right)+c^3\left(a-b\right)\)

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

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

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

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

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

\(=\left(a-b\right)\left(c-a\right)\left[\left(c-b\right)\left(b+c\right)+a\left(c-b\right)\right]\)

\(=-\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)\)

\(=a^3b-a^3c+b^3c-ab^3+c^3a-bc^3\)

\(=\left(a^3b-ab^3\right)+\left(a^3c-ac^3\right)+\left(b^3c-bc^3\right)\)

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

\(5x^2y+5xy^2-a^2x-a^2y\)

\(=5xy\left(x+y\right)-a^2\left(x+y\right)\)

\(=\left(x+y\right)\left(5xy-a^2\right)\)

`a^3-a^2 x -ay+xy`

`=a^2(a-x)-y(a-x)`

`=(a-x)(x^2-y)`

`x^2-2xy+x-2y`

`= (x^2+x)-(2xy+2y)`

`=x(x+1)-2y(x+1)`

`=(x+1)(x-2y)`

`x^2-2x+2y-xy`

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

`=x(x-2)-y(x-2)`

`=(x-2)(x-y)`

1,(a^2-y)(a-x)

2,(x+1)(x-2y)

a: \(x^2-4xy+4y^2-2x+4y-35\)

\(=\left(x^2-4xy+4y^2\right)-\left(2x-4y\right)-35\)

\(=\left(x-2y\right)^2-2\left(x-2y\right)-35\)

\(=\left(x-2y\right)^2-7\left(x-2y\right)+5\left(x-2y\right)-35\)

\(=\left(x-2y\right)\left(x-2y-7\right)+5\left(x-2y-7\right)\)

\(=\left(x-2y-7\right)\left(x-2y+5\right)\)

c: \(\left(xy+ab\right)^2+\left(ay-bx\right)^2\)

\(=x^2y^2+a^2b^2+2xyab+a^2y^2-2aybx+b^2x^2\)

\(=x^2y^2+a^2y^2+a^2b^2+b^2x^2\)

\(=y^2\left(x^2+a^2\right)+b^2\left(a^2+x^2\right)\)

\(=\left(x^2+a^2\right)\left(y^2+b^2\right)\)

a

=y(x+5)

b

=(2021+a)(x-y)