1) \(x^4-3x^2-x+3\)
\(=x\left(x^3-1\right)-3\left(x^2-1\right)\)
\(=x\left(x-1\right)\left(x^2+x+1\right)-3\left(x+1\right)\left(x-1\right)\)
\(=\left(x-1\right)\left(x^3+x^2+x-3x-3\right)\)
\(=\left(x-1\right)\left(x^3+x^2-2x-3\right)\)
2) \(3x+3y-x^2-2xy-y^2\)
\(=3\left(x+y\right)-\left(x^2+2xy+y^2\right)\)
\(=3\left(x+y\right)-\left(x+y\right)^2\)
\(=\left(x+y\right)\left(3-x-y\right)\)
3) \(x^4-x\)
\(=x\left(x^3-1\right)\)
\(=x\left(x-1\right)\left(x^2+x+1\right)\)
4) \(x^2+5x+4\)
\(=x^2+x+4x+4\)
\(=x\left(x+1\right)+4\left(x+1\right)\)
\(=\left(x+1\right)\left(x+4\right)\)
5) \(4x^2+4x-8\)
\(=4\left(x^2+x-2\right)\)
\(=4\left(x^2-x+2x-2\right)\)
\(=4\left[x\left(x-1\right)+2\left(x-1\right)\right]\)
\(=4\left(x-1\right)\left(x+2\right)\)
6) \(x^2+x-42\)
\(=x^2-6x+7x-42\)
\(=x\left(x-6\right)+7\left(x-6\right)\)
\(=\left(x-6\right)\left(x+7\right)\)
7) \(5x^2-3x-2\)
\(=5x^2-5x+2x-2\)
\(=5x\left(x-1\right)+2\left(x-1\right)\)
\(=\left(x-1\right)\left(5x+2\right)\)
8) \(x^2-4x-1\)
\(=\left(x^2-4x+4\right)-5\)
\(=\left(x-2\right)^2-5\)
\(=\left(x-2+\sqrt{5}\right)\left(x-2-\sqrt{5}\right)\)
9) \(x^2-7xy+12y^2\)
\(=x^2-3xy-4xy+12y^2\)
\(=x\left(x-3y\right)-4y\left(x-3y\right)\)
\(=\left(x-3y\right)\left(x-4y\right)\)
