For which values of b do the equations:
x^3 + bx^2 + 2bx - 1 = 0 and x^2 + (b-1)x + b = 0
have a common root?
I believe there's a little trick involved here:
Observe that is not a root of (2) for any choice of b. Hence this can not be a common root.
So we may replace the problem of finding b such that (3),(4) have a common root
Then a common root of (3),(4) must also be a root of the difference of (3),(4)
This implies that is the only possible common root they can have.
substituting in (2) gives 2 possible values for b.