Question 972180
x^3+2x^2+x-2.  I agree with synthetic division.  You've done everything right.  When this happens, I would graph it and see what the function looks like.

I graphed it and got a root about 0.7, and on the calculator 0.695.  That is very close to 2.  I find one root.  When I put the 0.695 back in, it is very close to a zero, within rounding error.

The derivative is 3x^2 +4x +1
That factors into (3x + 1) (x+1), so there are critical points at x=-1 and -(1/3), exactly as you describe, but they are a maximum (2nd derivative is negative for the first) and a minimum (second derivative positive for second), not zeros.  It is a cubic shape, as expected.  


{{{graph(300,300,-10,10,-10,10,x^3+2x^2+x-2)}}}