\(4x^2-9y^2+6x-9y\)

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

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

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

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

\(4x^2-9y^2+4x-6y\)

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

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

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

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

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

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

d)\(-\left(4x^2-12xy+9y^2\right)=-\left[\left(2x\right)^2-2×2x×3y+\left(3x\right)^2\right]=-\left[\left(2x-3y\right)^2\right]\)

\(8x^2+6x^3=2x^2\left(4+3x\right)\)

\(x^3-5x^2-4x+20=x^2\left(x-5\right)-4\left(x-5\right)=\left(x^2-4\right)\left(x-5\right)=\left(x-2\right)\left(x+2\right)\left(x-5\right)\)

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

a: \(8x^2+6x^3=2x^2\left(4+3x\right)\)

b: \(x^3-5x^2-4x+20\)

\(=x^2\left(x-5\right)-4\left(x-5\right)\)

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

c: \(x^2-4x+4-9y^2\)

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

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

a) \(9y^2-4x^2+6x+9y\)

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

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

b) \(x^3+4x^2-12x\)

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

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

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

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

a)\(9y^2-4x^2+6x+9y=\left(9y^2-4x^2\right)+\left(6x+9y\right)=\left[\left(3y\right)^2-\left(2x\right)^2\right]+3\left(2x+3y\right)\)

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

b)\(x^3+4x^2-12x=x^3-2x^2+6x^2-12x=\left(x^3-2x^2\right)+\left(6x^2-12x\right)\)

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

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

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

\(d,4x^2-6x-9y^2+9y=\left(4x^2-9y^2\right)-\left(6x-9y\right)=\left(2x-3y\right)\left(2x+3y\right)-3\left(2x-3y\right)=\left(2x+3y-3\right)\left(2x-3y\right)\)

ta có \(A=-\left(4x^2+9y^2+4x-6y-3\right)\)

              \(=-\left[\left(4x^2+4x+1\right)+\left(9y^2-6y+1\right)-5\right]\) 

                \(=-\left(2x+1\right)^2-\left(3y-1\right)^2+5\)

vì \(-\left(2x+1\right)^2< =0;-\left(3y-1\right)^2< =0\)

=> \(A< =5\)

               dấu = xảy ra <=> \(\hept{\begin{cases}x=-\frac{1}{2}\\y=\frac{1}{3}\end{cases}}\)

b)    ta có \(B=-\left(x^2-6x-5\right)=-\left[\left(x^2-6x+9\right)-14\right]\) 

                      \(=-\left(x-3\right)^2+14\)

mà \(-\left(x-3\right)^2< =0\) => b<=14

dấu = xảy ra  <=> \(x=3\)

`@` `\text {Ans}`

`\downarrow`

`4x^3 - 4x^2 - 9x + 9`

`= (4x^3 - 4x^2) - (9x - 9)`

`= 4x^2(x - 1) - 9(x - 1)`

`= (4x^2 - 9)(x - 1)`

____

`x^3 + 6x^2 + 11x + 6`

`= x^3 + x^2 + 5x^2 + 5x + 6x + 6`

`= (x^3 + x^2) + (5x^2 + 5x) + (6x + 6)`

`= x^2*(x + 1) + 5x(x + 1) + 6(x + 1)`

`= (x^2 + 5x + 6)(x+1)`

____

`x^2y - x^3 - 9y + 9x`

`= (x^2y - 9y) - (x^3 - 9x)`

`= y(x^2 - 9) - x(x^2 - 9)`

`= (y - x)(x^2 - 9)`

b: =x^3+x^2+5x^2+5x+6x+6

=(x+1)(x^2+5x+6)

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

c: =x^2(y-x)-9(y-x)

=(y-x)(x^2-9)

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

a: =(4x^3-4x^2)-(9x-9)

=4x^2(x-1)-9(x-1)

=(x-1)(4x^2-9)

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