a: \(\left(x^2+x\right)^2-2\cdot\left(x^2+x\right)-15\)
\(=\left(x^2+x\right)^2-5\left(x^2+x\right)+3\left(x^2+x\right)-15\)
\(=\left(x^2+x\right)\left(x^2+x-5\right)+3\left(x^2+x-5\right)\)
\(=\left(x^2+x-5\right)\left(x^2+x+3\right)\)
b: \(\left(x^2-2x\right)\left(x^2-2x-1\right)-6\)
\(=\left(x^2-2x\right)^2-\left(x^2-2x\right)-6\)
\(=\left(x^2-2x-3\right)\left(x^2-2x+2\right)=\left(x-3\right)\left(x+1\right)\left(x^2-2x+2\right)\)
c: (x-2)(x-3)(x-4)(x-5)+1
\(=\left(x^2-7x+10\right)\left(x^2-7x+12\right)+1\)
\(=\left(x^2-7x+11-1\right)\left(x^2-7x+11+1\right)+1\)
\(=\left(x^2-7x+11\right)^2-1+1=\left(x^2-7x+11\right)^2\)
d: \(x^2\left(x^4-1\right)\left(x^2+2\right)+1\)
\(=x^2\left(x^2-1\right)\left(x^2+1\right)\left(x^2+2\right)+1\)
\(=x^2\left(x^2+1\right)\left(x^2-1\right)\left(x^2+2\right)+1\)
\(=\left(x^4+x^2\right)\left(x^4+x^2-2\right)+1=\left(x^4+x^2\right)^2-2\left(x^4+x^2\right)+1\)
\(=\left(x^4+x^2-1\right)^2\)
e: \(4x^4-32x^2+1\)
\(=4x^4+4x^2+1-36x^2\)
\(=\left(2x^2+1\right)^2-\left(6x\right)^2=\left(2x^2-6x+1\right)\left(2x^2+6x+1\right)\)
f: \(x^8+14x^4+1\)
\(=x^8+2x^4+1-16x^4\)
\(=\left(x^4+1\right)^2-\left(4x^2\right)^2=\left(x^4+1-4x^2\right)\left(x^4+1+4x^2\right)\)
