a: \(4y+x^2y+4xy\)
\(=y\left(x^2+4x+4\right)\)
\(=y\left(x+2\right)^2\)
b: \(3x^2-6xy+3y^2-27z^2\)
\(=3\left(x^2-2xy+y^2-9z^2\right)\)
\(=3\left[\left(x^2-2xy+y^2\right)-9z^2\right]\)
\(=3\left[\left(x-y\right)^2-\left(3z\right)^2\right]\)
\(=3\left(x-y-3z\right)\left(x-y+3z\right)\)
c: \(4x-4y+x^2-2xy+y^2\)
\(=\left(4x-4y\right)+\left(x^2-2xy+y^2\right)\)
\(=4\left(x-y\right)+\left(x-y\right)^2\)
\(=\left(x-y\right)\left(x-y+4\right)\)
d: \(4x^2-25-\left(2x-5\right)\left(2x+7\right)\)
\(=\left(4x^2-25\right)-\left(2x-5\right)\left(2x+7\right)\)
\(=\left(2x-5\right)\left(2x+5\right)-\left(2x-5\right)\left(2x+7\right)\)
\(=\left(2x-5\right)\left(2x+5-2x-7\right)\)
\(=-2\cdot\left(2x-5\right)\)
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