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

x3+x2-x+2=x3+2x2-x2-2x+x+2=(x3+2x2)-(x2+2x)+(x+2)=x2(x+2)-x(x+2)+(x+2)=(x+2)(x2-x+1)

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

2x3-x2+5x+3=2x3+x2-2x2-x+6x+3=(2x3+x2)-(2x2+x)+(6x+3)=x2(2x+1)-x(2x+1)+3(2x+1)=(2x+1)(x2-x+3)

\(x^8+3x^4+4\)
\(=\left(x^8-x^6+2x^4\right)+\left(x^6-x^4+2x^2\right)+\left(2x^4-2x^2+4\right)\)
\(=x^4\left(x^4-x^2+2\right)+x^2\left(x^4-x^2+2\right)+2\left(x^4-x^2+2\right)\)
\(=\left(x^4+x^2+2\right)\left(x^4-x^2+2\right)\)
\(4x^4+4x^3+5x^2+2x+1\)
\(=\left(4x^4+2x^3+2x^2\right)+\left(2x^3+x^2+x\right)+\left(2x^2+x+1\right)\)
\(=2x^2\left(2x^2+x+1\right)+x\left(2x^2+x+1\right)+\left(2x^2+x+1\right)\)
\(=\left(2x^2+x+1\right)^2\)


\(2x^3-x^2+3x+6=2x^3-3x^2+6x+2x^2-3x+6\)
\(=x.\left(2x^2-3x+6\right)+\left(2x^2-3x+6\right)\)
\(=\left(2x^2-3x+6\right).\left(x+1\right)\)

\(x^3-2x^2-x\)
\(=x\cdot x^2-x\cdot2x-x\cdot1\)
\(=x\left(x^2-2x-1\right)\)