\(x^3-x^2-5x+125\)
\(=\left(x+5\right)\left(x^2-5x+25\right)-x\left(x+5\right)\)
\(=\left(x+5\right)\left(x^2-5x+25-x\right)\)
\(=\left(x+5\right)\left(x^2-6x+25\right)\)
\(x^6-x^4-9x^3+9x^2\)
\(=x^4\left(x^2-1\right)-9x^2\left(x-1\right)\)
\(=x^4\left(x-1\right)\left(x+1\right)-9x^2\left(x-1\right)\)
\(=x^2\left(x-1\right)\left[x^2\left(x+1\right)-9\right]\)
\(=x^2\left(x-1\right)\left(x^3+x^2-9\right)\)
\(x^4-4x^3+8x^2-16x+16\)
\(=\left(x^2+4\right)^2-4x\left(x^2+4\right)\)
\(=\left(x^2+4\right)\left(x^2+4-4x\right)\)
\(=\left(x^2+4\right)\left(x-2\right)^2\)
\(3a^2-6ab+3b^2-12c^2\)
\(=3\left(a^2-2ab+b^2-4c^2\right)\)
\(=3\left[\left(a-b\right)^2-\left(2c\right)^2\right]\)
\(=3\left(a-b+2c\right)\left(a-b-2c\right)\)
a) \(x^3-x^2-5x+125=\left(x+5\right)\left(x^2-5x+25\right)-x\left(x+5\right)=\left(x+5\right)\left(x^2-6x+25\right)\)
b) \(x^6-x^4-9x^3+9x^2=x^2\left(x^4-x^2-9x+9\right)=x^2\left[x^2\left(x-1\right)\left(x+1\right)-9\left(x-1\right)\right]=x^2\left(x-1\right)\left[x^2\left(x+1\right)-9\right]=x^2\left(x-1\right)\left(x^3+x^2-9\right)\)
c) \(x^4-4x^3+8x^2-16x+16=x^3\left(x-2\right)-2x^2\left(x-2\right)+4x\left(x-2\right)-8\left(x-2\right)=\left(x-2\right)\left(x^3-2x^2+4x+8\right)=\left(x-2\right)\left(x^2+4\right)\left(x-2\right)=\left(x-2\right)^2\left(x^2+4\right)\)
d) \(3a^2-6ab+3b^2-12c^2=3\left(a^2-2ab+b^2-4c^2\right)=3\left(a-b-2c\right)\left(a-b+2c\right)\)
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