Determine value of parameter, so that a polynomial has a root with multiplicity 2 I have the polynomial $P_a(x) = x^4 - x^2 + ax + 2$, where $a$ is a complex number, and I have to determine the values of $a$ such that $P_a(x)$ has a root with multiplicity at least two. I have tried two approaches, but none of them seems to work. Let's see:
We know that if $P_a(x)$ is to have a root with multiplicity at least two, then $P_a(x)$ and its derivative $P'_a(x)$ must have a common root. So, in the first approach I have computed the resultant of $P_a(x)$ and $P'_a(x)$ with the aid of Mathematica, which gives $512 - 83 a^2 - 8 a^4$. This resultant must be equal to zero, hence we obtain the following four solutions for $a$: 
$$
a = -\sqrt{-(83/16) + (37 \sqrt{17})/16}, \\
a =  \sqrt{-(83/16) + (37 \sqrt{17})/16}, \\
a = -(1/4) i \sqrt{ 83 + 37 \sqrt{17}}, \\
a = 1/4 i \sqrt{83 + 37 \sqrt{17}}
$$
In the second approach, I have computed the GCD of $P_a(x)$ and $P'_a(x)$, imposing the condition that the GCD must be at least linear. After successive divisions we get the remainder 
$$
\frac{1536 + 861 a^2 + 864 a^4}{3 (16 + 12 a^2)^2},
$$
which we want to become zero. Hence we obtain the following four solutions for $a$:
$$
a = -\sqrt{-(287/576) - (i \sqrt{507455})/576} \\
a = \sqrt{-(287/576) - (i \sqrt{507455})/576} \\
a = -\sqrt{-(287/576) + (i \sqrt{507455})/576} \\
a = \sqrt{-(287/576) + (i \sqrt{507455})/576}.   
$$
The first obvious problem is that both sets of solutions are different. Moreover, $a = 2$ should be a solution, since $x^4 - x^2 + 2x + 2$ has root $-1$ with multiplicity two, but this solution is not included in any of the above sets. Where is the error?  

Your resultant is false:
Resultant[$x^4 - x^2 + a x + 2$, $4 x^3 - 2 x + a$, $x$ ] = $1568 - 284\,a^2 - 27\,a^4$ whose 2 real roots are $a=2$ and $a=-2$.
Edit: It can be of interest to have a pictorial representation of what is going on ; here is a way: 
Equation $x^4 - x^2 + a x + 2=0$ can be considered as the equation verified by the abscissas of intersection point(s) of curve $y=x^4-x^2$ and straight line with equation $y=-a x - 2$. The desired values of $a$ correspond to the two tangency cases.

