Bài 5:
\(a,M=\left(2x-1\right)^2+2\left(2x-1\right)\left(3x+1\right)+\left(3x+1\right)^2\\ =\left[\left(2x-1\right)+\left(3x+1\right)\right]^2\\ =\left(2x-1+3x+1\right)^2\\ =\left(5x\right)^2\\ =\left(5\cdot-\dfrac{1}{5}\right)^2=\left(-1\right)^2=1\\ b,N=\left(3x-1\right)^2-2\left(9x^2-1\right)+\left(3x+1\right)^2\\ =\left(3x-1\right)^2-2\left(3x-1\right)\left(3x+1\right)+\left(3x+1\right)\\ =\left[\left(3x-1\right)-\left(3x+1\right)\right]^2\\ =\left(3x-1-3x-1\right)^2\\ =\left(-2\right)^2\\ =4\)
Bài 6:
\(a,P=27-27x+9x^2-x^3\\ =\left(3-x\right)^3\\ =\left[3-\left(-17\right)\right]^3\\ =20^3\\ =8000\\ b,Q=x^3+3x^2+3x\\ =\left(x^3+3x^2+3x+1\right)-1\\ =\left(x+1\right)^3-1\\ =\left(99+1\right)^3-1\\ =100^3-1\\ =1000000-1\\ =999999\)
Bài 4:
a: \(2x^3y+2xy^3+4x^2y^2-2xy\)
\(=2xy\cdot x^2+2xy\cdot y^2+2xy\cdot2xy-2xy\cdot1\)
\(=2xy\left(x^2+2xy+y^2-1\right)\)
\(=2xy\left[\left(x+y\right)^2-1\right]\)
\(=2xy\left(x+y+1\right)\left(x+y-1\right)\)
b: \(x^2+y^2-2xy+4x-4y\)
\(=\left(x^2-2xy+y^2\right)+4\left(x-y\right)\)
\(=\left(x-y\right)^2+4\left(x-y\right)\)
\(=\left(x-y\right)\left(x-y+4\right)\)
c: \(x^3-x+3x^2y+3xy^2+y^3-y\)
\(=\left(x^3+3x^2y+3xy^2+y^3\right)-\left(x+y\right)\)
\(=\left(x+y\right)^3-\left(x+y\right)\)
\(=\left(x+y\right)\left[\left(x+y\right)^2-1\right]=\left(x+y\right)\left(x+y+1\right)\left(x+y-1\right)\)
d: \(x^2-2xy+y^2-4z^2\)
\(=\left(x-y\right)^2-\left(2z\right)^2\)
\(=\left(x-y-2z\right)\left(x-y+2z\right)\)
e: \(x^2-x-y^2-y\)
\(=\left(x^2-y^2\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y\right)-\left(x+y\right)\)
\(=\left(x+y\right)\left(x-y-1\right)\)
f: \(x^2-2xy+y^2-z^2\)
\(=\left(x-y\right)^2-z^2\)
\(=\left(x-y-z\right)\left(x-y+z\right)\)