For any a, b , $\displaystyle \in R $ let a * b denote ab + a + b.
Show that if a* b = -1 then either a = -1 or b = -11.
Am not sure where to start, from here.
I assume that R means $\displaystyle \mathbb{R}$, the set of real numbers.
(If R stands for a generic ring, you'll need to specify that it's a domain.)
I also assume that your "b = -11" is a typo - that you intended "b = -1"
a*b = -1 implies ab + a + b = -1.
Now write that as "a, b stuff" = 0, and it will actually factor, and from there lead you to your result.