a ) \(x^3+3x^2+6x+4\)

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

\(=\left(x+1\right)^3+3\left(x+1\right)\)

\(=\left(x+1\right)\left[\left(x+1\right)^2+3\right]\)

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

b ) \(3a^2c^2+bd+3abc+acd\)

\(=\left(3a^2c^2+3abc\right)+\left(bd+acd\right)\)

\(=3ac\left(ac+b\right)+d\left(ac+b\right)\)

\(=\left(3ac+d\right)\left(ac+b\right)\)

c ) \(3a^2-6ab+3b^2-12c^2\)

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

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

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

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

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

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

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

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

\(=\left(y-1\right)\left[-x^2\left(y+1\right)+x+y\right]\)

\(=\left(y-1\right)\left[-x^2y-x^2+x+y\right]\)

\(=\left(y-1\right)\left[x\left(1-x\right)+y\left(1-x^2\right)\right]\)

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

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

e ) \(a^6-b^6=\left(a^3\right)^2-\left(b^3\right)^2=\left(a^3-b^3\right)\left(a^3+b^3\right)\) \(=\left(a-b\right)\left(a^2+ab+b^2\right)\left(a+b\right)\left(a^2-ab+b^2\right)\)