# Trouble factoring an expression

I've got the term $-1 + 2b^2 + 2b\sqrt {b^2-1}$ appearing several times within several larger expressions. It is easy to see that this expression factors into ${(b+\sqrt{b^2-1})^2}$, but when I use FullSimplify or Factor it doesn't change. Is there a way to make Mathematica factor this expression automatically? $b$ is defined to be real and $>1$.

term = -1 + 2 b^2 + 2 b Sqrt[b^2 - 1]

Indeed, the second expression, with a LeafCount of 13, is simpler than the first, with a LeafCount of 19.
Depending on the details of the actual problem you wish to solve, working with hyperbolic functions throughout instead of b might be preferable.