I can not understand this step that the teacher did and I do not understand how she applied the Taylor series and which of the formulas she used. I already searched the Internet but found no formula that would be useful to me.
Step: $$mc^2\bigg[1+\Big(\frac{\vec{p}c}{mc^2}\Big)^2\bigg]^{1/2}\approx mc^2\bigg[1+\frac{1}{2}\Big(\frac{\vec{p}c}{mc^2}\Big)^2\bigg]$$
The Taylor series of $(1+x)^n$ at $0$ is also known as the binomial series or generalized binomial theorem. It converges for $|x|<1$ and is given by
$$ (1+x)^n\approx 1+nx $$
Putting $n=\frac{1}{2}$ and $x=\left(\frac{pc}{mc^2}\right)^2$ gives the required result.