<s>[INST] Find all $a\in\mathbb{C}$ such that $f=X^4 −(a+4)X^3 +(4a+5)X^2 −(5a+2)X +2a$ has $a$ as a double root. Find all $a\in\mathbb{C}$ such that $f=X^4 −(a+4)X^3 +(4a+5)X^2 −(5a+2)X +2a$ has $a$ as a double root.

Here is what I did:
I know for a fact that all $f\in\mathbb{C}$ of degree $n\geq1$ have exactly $n$ roots. So, I write $f=(X-a)^2(X-b)(X-c)=X^4 - (2a+b+c)X^3 + (bc+2ac+2ab+a^2)X^2 - (2abc+a^2c+a^2b)+a^2bc$. I solved the system of equations ($2a+b+c=a+4$, ...) using Wolfram Alpha and found $\begin{cases} a=1,b=1,c=2 \\ a=1,b=2,c=1 \\ a=2,b=1,c=1\end{cases}$. Hence $a=2$ because when $a=1$, $a$ is a triple root.
I don't reckon this is the solution they were asking for. I mean, I wouldn't try to solve that system of equations without the help of a computer. So what could I do instead?
 [/INST] Since $f(a)=0$ and $f'(a)=(a-2)(a-1)^2$, $a$ is a multiple root if and only if $a=2$ and $a=1$. But $f''(a)=2 \left(3 a^2-8 a+5\right)$, and therefore $f''(1)=0$ and $f''(2)\ne0$. So, it's a double root if and only if $a=2$.
</s>