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\(\text{3x^2 - 22 xy - 4x + 8y + 7y^2 +1}\)
\(=3x^2-21xy-3x-x+y+7y+7y^2+1\)
\(=3x^2-21xy-3x-xy+7y^2+y-x+7y+1\)
\(=\left(3x^2-21xy-3x\right)-\left(xy-7y^2-y\right)-\left(x-7y-1\right)\)
\(=3x\left(x-7y-1\right)-y\left(x-7y-1\right)-\left(x-7y-1\right)\)
\(=\left(x-7y-1\right)\left(3x-y-1\right)\)
\(\left(x^2+xy\right)^2-\left(y^2+xy\right)^2\)
\(=\left(x^2+xy-y^2-xy\right)\left(x^2+xy+y^2+xy\right)\)
\(=\left(x^2-y^2\right)\left(x^2+2xy+y^2\right)\)
\(=\left(x-y\right)\left(x+y\right)\left(x+y\right)^2\)
\(=\left(x-y\right)\left(x+y\right)^3\)
•x3+y3+z3-3xyz=(x+y)3-3xy(x+y)+z3-3xyz
=(x+y+z)[(x+y)2-(x+y).z+z2]-3xy(x+y+z)
=(x+y+z)(x2+y2+z2+2xy-xz-yz) -3xy(x+y+z)
=(x+y+z)(x2+y2+z2-xy-yz-xz)
•(x2+xy)2-(y2+xy)2=[x(x+y)]2-[y(x+y)]2
=x2.(x+y)2-y2.(x+y)2
=(x+y)2.(x2-y2)=(x+y)2.(x+y).(x-y)
=(x+y)3(x-y)
•3x2-3x-36=3.(x2-x-12)
=3(x2-4x+3x-12)
=3[x(x-4)+3(x-4)]=3(x-4)(x+3)
D = x2 - 4x - y2 - 8y - 12
= (x2 - 4x + 4) - (y2 + 8y + 16)
= (x - 2)2 - (y + 4)2
= (x + y + 2)(x - y - 6)
\(D=x^2-4x-y^2-8y-12\)
\(=x^2-4x-y^2-8y+4-16\)
\(=\left(x^2-4x+4\right)-\left(y^2+8y+16\right)\)
\(=\left(x-2\right)^2-\left(y+4\right)^2\)
\(=\left(x-2-y-4\right)\left(x-2+y+4\right)\)
\(=\left(x-y-6\right)\left(x+y+2\right)\)
x2 - 2xy - 8y2 = ( x2 - 2xy + y2 ) - 9y2 = ( x - y )2 - ( 3y )2 ( x - y - 3y ) ( x - y + 3y )
\(x^2+y^2-x^2y^2+xy-x-y\)
\(=x^2-x^2y^2+y^2-y+xy-x\)
\(=x^2\left(1-y^2\right)+y\left(y-1\right)+x\left(y-1\right)\)
\(=x^2\left(1-y\right)\left(y+1\right)+y\left(y-1\right)+x\left(y-1\right)\)
\(=\left(y-1\right)\left[-x^2\left(y+1\right)+y-x\right]\)
\(=\left(y-1\right)\left[-x^2y-x^2+y-x\right]\)