a) x - 2y + x2 - 4y2
= x - 2y + ( x - 2y).( x+2y)
= (x - 2y).(1+ x + 2y)
b) x2- 4x2y2 + y2 + 2xy
= (x + y)2 - 4x2y2
= (x + y -2xy).(x + y + 2xy)
c) x6 - x4 + 2x3 + 2x2
= x2.(x4 - x2 + 2x + 2)
= x2.[x2.(x2 - 1) + 2(x+1)]
= x2.[x2.(x-1).(x+1) + 2(x+1)]
= x2.(x+1).(x2.(x-1)+2)
= x2.(x+1).(x3 - x2 + 2)
= x2.(x+1).(x3+x2-2x2+2)
= x2.(x+1).[x2( x+1) +2 (-x2 +1)]
= x2.(x+1).[x2( x+1) +2 (1+x).(1-x)]
= x2.(x+1)2.(x2 + 2 -2x)
d) x3 + 3x2 + 3x + 1- 8y3
= (x+1)3 - 8y3
= ( x+1 - 2y).[(x+1)2 + (x+1). 2y + 4y2]
= ( x + 1 - 2y).(x2 + 2x + 1 + 2xy + 2y + 4y2)
a/ \(x-2y+x^2-4y^2\)
\(=\left(x-2y\right)+x^2-\left(2y\right)^2\)
\(=\left(x-2y\right)+\left(x-2y\right)\left(x+2y\right)\)
\(=\left(x-2y\right)\left[1+\left(x+2y\right)\right]\)
\(=\left(x-2y\right)\left(1+x+2y\right)\)
b/ \(x^2-4x^2y^2+y^2+2xy\)
\(=\left(x^2+2xy+y^2\right)-4x^2y^2\)
\(=\left(x+y\right)^2-\left(2xy\right)^2\)
\(=\left(x+y-2xy\right)\left(x+y+2xy\right)\)
c/ \(x^6-x^4+2x^3+2x^2\)
\(=x^4\left(x^2-1\right)+2x^2\left(x+1\right)\)
\(=x^4\left(x-1\right)\left(x+1\right)+2x^2\left(x+1\right)\)
\(=\left(x+1\right)\left[x^4\left(x-1\right)+2x^2\right]\)
d/ \(x^3+3x^2+3x+1-8y^3\)
\(=\left(x+1\right)^3-\left(2y\right)^3\)
\(=\left(x+1-2y\right)\left[\left(x+1\right)^2+\left(x+1\right)2y+\left(2y\right)^2\right]\)
\(=\left(x+1-2y\right)\left[\left(x^2+2x+1\right)+2xy+2y+4y^2\right]\)
\(=\left(x+1-2y\right)\left(x^2+2x+1+2xy+2y+4y^2\right)\)
