Find the cubic equation of roots $α, β, γ$. Taken from Fitzpatrick $4$ unit course textbook. The question says:
If the cubic equation $\ ax^3+bx^2+cx+d$ has roots $α, β, γ$. Find the cubic equation who's roots are $α^2, β^2, γ^2$
I keep getting a $±$ sign that I can't get rid of. The answer in the back is $x(ax+c)^2=(bx+d)^2$
Thanks for any help.

$$ax^3+bx^2+cx+d=a^3(x-\alpha)(x-\beta)(x-\gamma)$$
Set $x=y$ to get $$ay^3+by^2+cy+d=a^3(y-\alpha)(y-\beta)(y-\gamma)\ \ \ \  (1)$$
Set $x=-y$ to get $$a(-y)^3+b(-y)^2+c(-y)+d=a^3(-y-\alpha)(-y-\beta)(-y-\gamma)$$
$$\iff ay^3-by^2+cy-d=a^3(y+\alpha)(y+\beta)(y+\gamma)\ \ \ \ (2)$$
Multiply $(1),(2)$ to get $$a^6(y^2-\alpha^2)(y^2-\beta^2)(y^2-\gamma^2)=(ay^3+cy)^2-(by^2+d)^2$$
Set $y^2=u$
