The Step "Place under the common denominator", the $-1$ turned into $-e$, but it originally was $-r$. I understand with factoring out the $r$, it becomes $-1$, but how did it become a $-e$?
Can you please specifically answer to the question and not in an abstract manner?
$$ \begin{array}{ll} \displaystyle\frac{E}{e}=\frac{R+r}{r}&\\ \displaystyle\frac{E}{e}\times r=\frac{R+r}{r}\times r&\text{multiply both sides by $r$}\\ \displaystyle\frac{r\;E}{e}=R+r&\text{simplify}\\ \displaystyle\frac{r\;E}{e}-r=R&\text{substract $r$ from both sides}\\ \displaystyle r\left(\frac{E}{e}-1\right)=R&\text{factor out $r$}\\ \displaystyle\color{red}{r\left(\frac{E-e}{e}\right)=R}&\color{red}{\text{place under common denominator}}\\ \displaystyle r\left(\frac{E-e}{e}\right)\times\frac{e}{E-e}=R\times\frac{e}{E-e}&\text{multiply both sides by $\displaystyle\frac{e}{E-e}$}\\ \displaystyle r=\frac{eR}{E-e} \end{array} $$
$$ r\left(\frac{E}{e} -1 \right) = R$$ $$ r\left(\frac{E}{e} -1\times \frac{e}{e} \right) = r\left(\frac{E - e}{e} \right)$$
You can multiply anything by $1$ and $e$ divided by $e$ is $1$. Once you have two fractions with the same denominator, you can make the denominator common:
$$\frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}$$