Prove by induction that $n(n+1)(n+5)$ is multiple of $3.$ $n(n+1)(n+5) = 3d$
I cannot figure out how to solve this homework question. A friend gave me a solution I couldn't make sense of, and I hope there's something easier out there. Also, what would be the general approach towards questions of this form?

Since $$\begin{align}(n+1)(n+2)(n+6)-n(n+1)(n+5)&=(n+1)\left\{(n^2+8n+12)-(n^2+5n)\right\}\\&=(n+1)(3n+12)\\&=3(n+1)(n+4)\end{align} $$
we have
$$(n+1)(n+2)(n+6)=n(n+1)(n+5)+3(n+1)(n+4).$$
