Prove that polynomial doesn't admit a particular solution

Question

Prove that the equation$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$ does not admit $$x=-2$$ as a triple root.

Answer 1

Check when is $f(2)$, $f'(2)$ and $f''(2)$ equal zero.

Answer 2

Another approach.

Suppose it did. Let the other root be $r$. Use Vieta's formulae to compute $r$ and $a$.