`(ax+by)^2-(ay+bx)^2`

`=(ax+by+ay+bx)(ax+by-ay-bx)`

`=[a(x+y)+b(x+y)][a(x-y)-b(x-y)]`

`=(x+y)(a+b)(x-y)(a-b)`

\(\left(ax+by\right)^2-\left(ay+bx\right)^2\)

\(=\left(ax-ay-bx+by\right)\left(ax+ay+bx+by\right)\)

\(=\left(x-y\right)\left(a-b\right)\left(x+y\right)\left(a+b\right)\)

\(3x^2-7x+4=\left(3x^2-3x\right)-\left(4x-4\right)=3x\left(x-1\right)-4\left(x-1\right)=\left(x-1\right)\left(3x-4\right)\)

3x^2 -7x+4

= 3x^2 -3x-4x+4

= 3x ( x-1) -4(x-1)

= (3x-4)(x-1)

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

x³ - 9x² + 7x + 1

= x³ - x2 - 8x² + 8x - x + 1

= (x³ - x²) - (8x² - 8x) - (x - 1)

= x²(x - 1) - 8x(x - 1) - (x - 1)

= (x - 1)(x² - 8x -1)

\(x^2+2xy+7x+7y+y^2+10\)

\(=\left(x^2+2xy+y^2\right)+\left(7x+7y\right)+\frac{49}{4}-\frac{9}{4}\)

\(=\left(x+y\right)^2+7\left(x+y\right)+\frac{49}{4}-\frac{9}{4}\)

\(=\left(x+y+\frac{7}{2}\right)^2-\frac{9}{4}\)

\(=\left(x+y+\frac{7}{2}-\frac{3}{2}\right)\left(x+y+\frac{7}{2}+\frac{3}{2}\right)\)

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

b)Ta có: x²y+xy²+x+y=2010

<=>xy.x+xy.y+x+y=2010

<=>11x+11y+x+y=2010

<=>12(x+y)=2010

<=>x+y=167,5

=>(x+y)²=28056,25

<=>x²+y²+2xy=28056,25

<=>x²+y²=28034,25

\(\left(x+y\right)^2+7\left(x+y\right)+\frac{49}{4}-\frac{9}{4}\)

\(=\left(x+y\right)^2+2\left(x+y\right).\frac{7}{2}+\left(\frac{7}{2}\right)^2-\frac{9}{4}\)

\(=\left(x+y+\frac{7}{2}\right)^2-\frac{9}{4}\)   (áp dụng HĐT số 1)

\(=\left(x+y+\frac{7}{2}\right)^2-\left(\frac{3}{2}\right)^2\)  

\(=\left(x+y+\frac{7}{2}-\frac{3}{2}\right)\left(x+y+\frac{7}{2}+\frac{3}{2}\right)\)(áp dụng HĐT số 3)

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

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2x^4 - 2x^3 +9x^3 - 9x^2 +7x^2 -  7x - 6x + 6 = 2x^3(x-1)+9x^2(x-1)+7x(x-1)-6(x-1) = (x-1)(2x^3+9x^2+7x-6)                                               =(x-1)(x+2)(x+3)(2x-1)

=3x(x+2y)

=3x(x+2y)

3x² + 6xy

= 3x(x+2y)

\(2x^2+x-6\)

\(=2x^2-3x+4x-6\)

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

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

Tham Khảo

2x² + x - 6

= 2x² + 4x - 3x - 6

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

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