\(a=n\left(n+1\right)\left(n+2\right)\left(n+3\right)+1\)
\(a=\left[n\left(n+3\right)\right]\left[\left(n+2\right)\left(n+1\right)\right]+1\)
\(a=\left(n^2+3n\right)\left(n^2+n+2n+2\right)+1\)
\(a=\left(n^2+3n\right)\left(n^2+3n+2\right)+1\)
\(a=\left(n^2+3n+1-1\right)\left(n^2+3n+1+1\right)+1\)
\(a=\left(n^2+3n+1\right)^2-1+1\)
\(a=\left(n^2+3n+1\right)^2\)
Vì \(n\in Z\) nên \(n^2+3n+1\in Z\)
\(a\) là số chính phương (đpcm)