1. ## common root

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?

2. I believe there's a little trick involved here:

Let
(1) $x^3+bx^2+2bx-1=0$
(2) $x^2+(b-1)x+b=0$

Observe that $x=-1$ 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

(3) $x^3+bx^2+2bx-1=0$
(4) $(x+1)(x^2+(b-1)x+b)=0$

Then a common root of (3),(4) must also be a root of the difference of (3),(4)

$(3)-(4)= x^3+bx^2+2bx-1 - (x+1)(x^2+(b-1)x+b) = x-(1+b)$

This implies that $x=1+b$ is the only possible common root they can have.

substituting $x= 1+b$ in (2) gives 2 possible values for b.