Factor A Degree 100 Polynomial (III)

Algebra Level 5

Can the polynomial \[ (x-1)(x-2)(x-3) \cdots (x-100) + 1 + 101!\] be factorized into 2 non-constant polynomials with integer coefficients?

\[\] Note: \(101! = 1\times2\times\cdots\times101\).