a) x² - 2xy + y² - 4m² + 4mn - n² = (x - y)² - [(2m)² -  2.2m.n + n²] = (x - y)² - (2m - n)²

= [(x - y) - (2m - n)][(x - y) + (2m - n)] = (x - y - 2m + n)(x - y + 2m - n)

b) x² - 4x²y² + y² + 2xy = x² + 2xy + y² - 4x²y² = (x + y)² - (2xy)² = (x + y - 2xy)(x + y + 2xy)

c) x⁶ - y⁶ = (x³)² - (y³)² = (x³ - y³)(x³ + y³) = (x - y)(x² + xy + y²)(x + y)(x² - xy - y²)

d) 25 - a² + 2ab - b² = 25 - (a² - 2ab + b²) = 5² - (a - b)² = (5 - a + b)(5 + a - b)

xin lỗi các bạn, đề mink có vấn đề: ý c phải là: x^6 - y^6

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

a) Ta có: \(\left(4x^2-3x-18\right)^2-\left(4x^2+3x\right)^2\)

\(=\left(4x^2-3x-18-4x^2-3x\right)\left(4x^2-3x-18+4x^2+3x\right)\)

\(=\left(-6x-18\right)\left(8x^2-18\right)\)

\(=-6\left(x+3\right)\cdot2\left(4x^2-9\right)\)

\(=-12\left(x+3\right)\left(2x-3\right)\left(2x+3\right)\)

b) Ta có: \(9\left(x+y-1\right)^2-4\left(2x+3y+1\right)^2\)

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

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

\(=-\left(x+3y+5\right)\left(7x+9y-1\right)\)

c) Ta có: \(-4x^2+12xy-9y^2+25\)

\(=-\left(4x^2-12xy+9y^2-25\right)\)

\(=-\left[\left(2x-3y\right)^2-25\right]\)

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

d) Ta có: \(x^2-2xy+y^2-4m^2+4mn-n^2\)

\(=\left(x^2-2xy+y^2\right)-\left(4m^2-4mn+n^2\right)\)

\(=\left(x-y\right)^2-\left(2m-n\right)^2\)

\(=\left(x-y-2m+n\right)\left(x-y+2m-n\right)\)

a) (4x²-3x-18)²-(4x²+3x)²

=(4x²-3x-18-4x²-3x)(4x²-3x-18+4x²+3x)

=(-6x-18)(8x²-18)

=-48x³+108x-144x²+324

a) \(x^2-xy+x-y\)

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

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

b) \(x^2+5x+6\)

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

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

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

\(2xy-x^2-y^2+16\)

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

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

a) ( 4x² - 3x - 18 )² - ( 4x² + 3x )²

= [ ( 4x² - 3x - 18 ) - ( 4x² + 3x ) ][ ( 4x² - 3x - 18 ) + ( 4x² + 3x ) ]

= ( 4x² - 3x - 18 - 4x² - 3x )( 4x² - 3x - 18 + 4x² + 3x )

= ( -6x - 18 )( 8x² - 18 )

= -6( x + 3 ).2( 4x² - 9 )

= -12( x + 3 )( 2x - 3 )( 2x + 3 )

b) 9( x + y - 1 )² - 4( 2x + 3y + 1 )²

= 3²( x + y - 1 )² - 2²( 2x + 3y + 1 )²

= [ 3( x + y - 1 ) ]² - [ 2( 2x + 3y + 1 ) ]²

= ( 3x + 3y - 3 )² - ( 4x + 6y + 2 )²

= [ ( 3x + 3y - 3 ) - ( 4x + 6y + 2 ) ][ ( 3x + 3y - 3 ) + ( 4x + 6y + 2 ) ]

= ( 3x + 3y - 3 - 4x - 6y - 2 )( 3x + 3y - 3 + 4x + 6y + 2 )

= ( -x - 3y - 5 )( 7x + 9y - 1 )

c) -4x² + 12xy - 9y² + 25

= 25 - ( 4x² - 12xy + 9y² )

= 5² - ( 2x - 3y )²

= [ 5 - ( 2x - 3y ) ][ 5 + ( 2x - 3y ) ]

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

d) x² - 2xy + y² - 4m² + 4mn - n²

= ( x² - 2xy + y² ) - ( 4m² - 4mn + n² )

= ( x - y )² - ( 2m - n )²

= [ ( x - y ) - ( 2m - n ) ][ ( x - y ) + ( 2m - n ) ]

= ( x - y - 2m + n )( x - y + 2m - n )

a) (a+b+c)^2 + (a+b-c)^2 - 4c^2

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

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

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

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

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

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

b) 4a^2b^2 - (a^2+b^2-c^2)^2

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

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

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

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

c) a(b^3-c^3) + b(c^3-a^3) + c(a^3-b^3)

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

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

\(=a^3\left(c-b\right)+\left(c-b\right)\left(bc^2+b^2c\right)-\left(c-b\right)\left(ac^2+abc+ab^2\right)\)

\(=\left(c-b\right)\left(a^3+bc^2+b^2c-ac^2-abc-ab^2\right)\)

a) (a+b+c)^2 + (a+b-c)^2 - 4c^2

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

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

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

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

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

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

b) 4a^2b^2 - (a^2+b^2-c^2)^2

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

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

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

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

c) a(b^3-c^3) + b(c^3-a^3) + c(a^3-b^3)

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

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

\(=a^3\left(c-b\right)+\left(c-b\right)\left(bc^2+b^2c\right)-\left(c-b\right)\left(ac^2+abc+ab^2\right)\)

\(=\left(c-b\right)\left(a^3+bc^2+b^2c-ac^2-abc-ab^2\right)\)

 Kai Parker

`a. =4(x^2+4x+3)=4(x^2+3x+x+3)=4(x+3)(x+1)`

`b. =x^2+8x-7x-56=x(x+8)-7(x+8)=(x+8)(x-7)`

`c. =x^2-9x+8x-72=x(x-9)+8(x-9)=(x-9)(x+8)`

`d. =(x-y)^2-9=(x-y-3)(x-y+3)`