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b: \(=x^2-4x+4-y^2-6y-9\)
\(=\left(x-2\right)^2-\left(y+3\right)^2\)
\(=\left(x-2-y-3\right)\left(x-2+y+3\right)\)
\(=\left(x-y-5\right)\left(x+y+1\right)\)
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\(x^3-3x^2+1-3x=\left(x^3+1\right)-3x^2-3x\)
\(=\left(x+1\right)\left(x^2-x+1\right)-3x\left(x+1\right)=\left(x+1\right)\left(x^2-x+1-3x\right)=\left(x+1\right)\left(x^2-4x+1\right)\)
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\(x^2+7x+12=x\left(x+3\right)+4\left(x+3\right)=\left(x+3\right)\left(x+4\right)\)
\(=x^2+3x+4x+12\)
\(=x\left(x+3\right)+4\left(x+3\right)\)
\(=\left(x+3\right)\left(x+4\right)\)
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\(x^8+3x^3+1\)
\(=x^8-x^4+4x^4+4\)
\(=\left(x^4-1\right)\cdot\left(x^4+1\right)+4\cdot\left(x^4+1\right)\)
\(=\left(x^4+1\right)\cdot\left(x^4-1+4\right)\)
\(=\left(x^4+1\right)\cdot\left(x^4+3\right)\)
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\(=x^2\left(x+3\right)-4\left(x+3\right)\)
\(=\left(x+3\right)\left(x^2-4\right)\)
\(=\left(x+3\right)\left(x-2\right)\left(x+2\right)\)
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x3-3x2-4x+12=(x3-3x2)-(4x-12)=x2(x-3)-4(x-3)=(x-3)(x2-4)=(x-3)(x-2)(x+2)
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\(x^3-3x^2-4x+12\)
\(=\left(x^3-3x^2\right)-\left(4x-12\right)\)
\(=x^2\left(x-3\right)-4\left(x-3\right)\)
\(=\left(x-3\right)\left(x^2-4\right)\)
\(=\left(x-3\right)\left(x-2\right)\left(x+2\right)\)