Help me to understand the part of the derivation of posterior distribution hyperparameters

Question

I've looked on how to find hyper parameters of posterior distribution for normal distribution likelihood with unknown mean and precision.

Here is a derivation described https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf

Im trying to understand how it is done.

I've been looking on some likelihood equation derivation (61 equation in the paper above). I was following on, but I couldn't figure out how one transformation is done.

Can you help me with one part, please? How comes this:

equation form 1

becomes this

equation form 2

the full equation if you like equation

Answer 1

again. I've spent almost half a day to solve this puzzle and transformed this equation to the same form as in the book.

It turns out that the trick of adding -2ab + 2ab to equation sum did it, I was able to collect vars to the form of (a-b)^2 and solved it to the final form with the mean x.