a) \(\left(3x-1\right)^2-16=\left(3x-1\right)^2-4^2=\left(3x-1-4\right)\left(3x-1+4\right)=\left(3x-5\right)\left(3x+3\right)\)b) \(\left(2x+5\right)^2-\left(x-9\right)^2=\left(2x+5+x-9\right)\left(2x+5-x+9\right)=\left(3x-4\right)\left(x+14\right)\)
c) \(\left(y^3+8\right)+\left(y^2-4\right)\)
\(=\left(y+2\right)\left(y^2-2y+4\right)+\left(y-2\right)\left(y+2\right)\)
\(=\left(y+2\right)\left(y^2-2y+4+y-2\right)\)
\(=\left(y+2\right)\left(y^2-y+2\right)\)
d) \(x^4-1=\left(x^2\right)^2-1^2=\left(x^2-1\right)\left(x^2+1\right)=\left(x-1\right)\left(x+1\right)\left(x^2+1\right)\)
e) \(81y^2-\left(y^2+6y\right)^2\)
\(=\left(9y\right)^2-\left(y^2+6y\right)^2\)
\(=\left(9y-y^2-6y\right)\left(9y+y^2+6y\right)\)
\(=\left(3y-y^2\right)\left(15y+y^2\right)\)