a: \(5x\left(x-1\right)-3x\left(x-1\right)\)
\(=\left(x-1\right)\left(5x-3x\right)\)
\(=2x\cdot\left(x-1\right)\)
b: \(x^2-9\)
\(=x^2-3^2\)
\(=\left(x-3\right)\left(x+3\right)\)
c: \(4x^2-25\)
\(=\left(2x\right)^2-5^2\)
\(=\left(2x-5\right)\left(2x+5\right)\)
d: \(x^6-y^6\)
\(=\left(x^3-y^3\right)\left(x^3+y^3\right)\)
\(=\left(x-y\right)\left(x+y\right)\left(x^2-xy+y^2\right)\left(x^2+xy+y^2\right)\)
e: \(9x^2+6xy+y^2\)
\(=\left(3x\right)^2+2\cdot3x\cdot y+y^2\)
\(=\left(3x+y\right)^2\)
g: \(6x-9-x^2\)
\(=-\left(x^2-6x+9\right)\)
\(=-\left(x-3\right)^2\)
f: \(x^2+4y^2+4xy\)
\(=x^2+2\cdot x\cdot2y+\left(2y\right)^2\)
\(=\left(x+2y\right)^2\)
h: \(\left(5x+1\right)^2-\left(x+1\right)^2\)
\(=\left(5x+1+x+1\right)\left(5x+1-x-1\right)\)
\(=\left(6x+2\right)\cdot4x\)
\(=4x\cdot2\cdot\left(3x+1\right)=8x\left(3x+1\right)\)
i: \(\left(x+y\right)^2-\left(x-y\right)^2\)
\(=\left(x+y-x+y\right)\left(x+y+x-y\right)\)
\(=2y\cdot2x=4xy\)
