Solving an 8th order polynomial in terms of unknown variables

This may be far fetched, but I have an 8th order polynomial which I would like to solve. The polynomial is expressed in terms of variables: $A, B, C, D$. From the context of the problem I know that the variables are greater than 0 and there are two real solutions and then the other 6 solutions are complex.

Is there a way to solve (using computer software) such a polynomial? For reference this is the polynomial I am referring to: $$ f(x)=Ax^8-2Dx^6+(Bx-Cx^3)^2 $$

From a comment, it would work to let $f(x)=g(x^2)$, where $g(x)=Ax^4−2𝐷𝑥^3+𝑥(𝐵−𝐶𝑥)^2$. How can this, now fourth order, polynomial be solved.

It is not bad since, expanding, you have $$f(x)= A x^2 \Big[\frac{B^2}{A}-\frac{2 B C }{A}x^2+\frac{ C^2-2 D}{A}x^4+x^6 \Big]$$ Let $$x^2=y \qquad a=\frac{ C^2-2 D}{A}\qquad b=-\frac{2 B C }{A}\qquad c=\frac{B^2}{A}$$ and you need to solve the cubic $$y^3+a y^2+b y+c=0$$

Follow the steps given here and use, depending of the sign of $\Delta$, either the trigonometric or the hyperbolic method. This will give much nicer looking formula than Cardano method.