Proof: For every positive integer $n$, there is a sequence of $n$ consecutive positive integers containing no primes.

Question

Let $x=(n+1)!+2$. I get how to prove that $x$ or $x+1$ is prime, but there is a step in my book that proves that $x+i$ is prime like this:

$x+i=(1)(2)(3)(4)....(n+1)+(i+2)$.

But then it factors out to this:

$x+i=(1)(2)(3)(i+1)(i+3)...(n+1)+1$.

How does $i+2$ factor out to this?