Finding the roots and the rescaling of an equation

Question

This question is taken from Hinch's book on perturbation.

I need to find the rescalings $x=\delta X$ and the roots of the equation $\epsilon^2x^3+x^2+2x+\epsilon=0$

I have found to possible rescalings $\delta$~$1 $ and $\delta$~$\epsilon$ and therefore found two possible roots

$x_1=-\frac{1}{2}\epsilon-\frac{1}{8}\epsilon^{2}+o(\epsilon^{3})$

$x_2=-2+0.5\epsilon+o(\epsilon^{2})$

However I'm struggeling to find the third one.

Thanks

Answer 1

The third variant is $δ\sim ϵ^{-2}$ so that the polynomial reads as $$ X^3+X^2+ϵ^2X+ϵ^5=0 $$ with a solution starting with $$ X=-\frac{1+\sqrt{1-4ϵ^2}}2+O(ϵ^5)\iff x=-\frac{1+\sqrt{1-4ϵ^2}}{2ϵ^2}+O(ϵ^3) $$

Answer 2

Once you find $2$ roots, make use of the fact that sum of the roots is $-\dfrac1{\epsilon^2}$ or the fact that the product of the roots is $-\dfrac1{\epsilon}$.