a) \(27+27x+9x^2+x^3\)
\(=\left(x^3+27\right)+\left(27x+9x^2\right)\)
\(=\left(x+3\right)\left(x^2-3x+9\right)+9x\left(x+3\right)\)
\(=\left(x+3\right)\left(x^2+6x+9\right)\)
\(=\left(x+3\right)^3\)
b) \(5x^2-10xy+5y^2-20z^2\)
\(=\left(\sqrt{5}x-\sqrt{5}y\right)^2-20z^2\)
\(=\left(\sqrt{5}x-\sqrt{5}y+2\sqrt{5}z\right)\left(\sqrt{5}x-\sqrt{5}y-2\sqrt{5}z\right)\)
c) \(\left(x^2+1\right)^2-4x^2\)
\(=\left(x^2+2x+1\right)\left(x^2-2x+1\right)\)
\(=\left(x+1\right)^2\left(x-1\right)^2\)
\(=\left[\left(x+1\right)\left(x-1\right)\right]^2\)
\(=\left(x^2-1\right)^2\)
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