The polynomial that I consider is as follows:
$$ x^{4} + 12(2\epsilon - 1) x^{2} = 0 $$
I use regular perturbation which is as follows:
$$ x = x_{0} + \epsilon x_{1} + \epsilon^{2} x_{2} $$ I substitute this expression into the equation and found that $x_{0} = 0 (\text{ repeated}), 2 \sqrt{3}, -2\sqrt{3}$. The last two distinct roots can give me $x^{(3,4)} = \pm 2 \sqrt{3} \pm (2\sqrt{3}) \epsilon \pm \epsilon^{2} \sqrt{3}$. However, I got some difficulties in finding the correction to the repeated roots $x_{0} = 0$ (which will gives $x_{1}, x_{2} =0$ in the expression). My attempt is that I try to expand this by rescaling as $x = \epsilon^{\alpha} X$ where $\alpha > 0$.
$$ \epsilon^{4\alpha} X^{4} + 12 (2 \epsilon - 1) \epsilon^{2\alpha} X^{2} = 0 $$
How can I balance this equation so that I can find the correction to the repeated root at $x = 0$?