Expression factoring question

This is from a simple book explaining differentiation to the uninitiated and I don't understand the factoring. Can anyone help me understand how equation 3 is derived? Thanks

Let $y = x^{-2}$

Then proceed:

$y + dy = (x + dx)^{-2}$

$y + dy = x^{-2} (1 + dx/x)^{-2}$

Using simple algebraic manipulations you have $$(x+dx)^2=\left(x\left(1+\frac{dx}{x}\right)\right)^2 =x^2\left(1+\frac{dx}{x}\right)^2$$ and therefore $$(x+dx)^{-2}=x^{-2}\left(1+\frac{dx}{x}\right)^{-2}$$ but note that this is no justification about any differential properties, that may be the goal, only pure algebra for the RHS.