In the book "What is Mathematics?" there is a section that provides an inductive proof of the arithmetic progression.
Part of this proof is:
$\frac{r(r+1)+2(r+1)}{2}=\frac{(r+1)(r+2)}{2}$
I don't understand how you end up with the right hand side from the left hand side. Using the distributive law, I can come up with the following:
$\frac{r(r+1)+2(r+1)}{2}=\frac{(r^2+r)+(2r+1)}{2}$
But that seems a million miles away from $\frac{(r+1)(r+2)}{2}$.
Judging from the question and the comments you are not used to factoring common terms:
The distributive law of multiplication over addition tells us that $a(b+c)=ab+ac$ (as presented in the second page of the book)
Let $a=r+1$, $b=r$ and $c=2$, $(r+2)(r+1)$ is then equal to $r(r+1)+2(r+1)$.
Generally speaking when you have the sum of different expressions with a common term you can factor it out