Simplifying the first part of the quaternion rotation formula for a matrix.

Question

In the image below, I have the simplified part in green box. Why is there a $2$. How does it simplify from $(q_4^2 - q_1^2-q_2^2-q_3^2)$ to $2q_4^2 - 1$

Here Image here

Answer 1

The first line says that $$q_4^2-1=-q_1^2-q_2^2-q_3^2$$and they are just inserting that into$$q_4^2+( -q_1^2-q_2^2-q_3^2)=q_4^2+(q_4^2-1)$$