Making up terms

A powerful algebra move is to make up terms: add and subtract the same quantity, or split a term into two, so the expression keeps its value but takes a form you can actually work with.
Because you add and subtract the same amount, you are really adding $0$, so nothing changes.

One use is completing the square.
To reshape $x^2 + 4x$, make up the terms $+4$ and $-4$: $x^2 + 4x + 4 - 4$.
The first three terms form a perfect square, so this becomes $(x + 2)^2 - 4$, which is far easier to graph or solve.

Another use is splitting a fraction.
To simplify $\frac{x}{x + 5}$, rewrite the top $x$ as $(x + 5) - 5$: $\frac{(x + 5) - 5}{x + 5}$.
Splitting the fraction gives $1 - \frac{5}{x + 5}$, a form that makes the behaviour and the asymptote obvious.

So decide what form you want — a perfect square, or a proper split — then work out the term you need and add it and subtract it together.
Making up a term is legal precisely because the added and subtracted parts cancel.
It turns an awkward expression into one you can graph, simplify, or solve.

Rewrite $x^2 + 6x$ by completing the square.
Make up $+9$ and $-9$: $x^2 + 6x + 9 - 9$.
The first three terms are a perfect square, so this is $(x + 3)^2 - 9$.

Rewrite $\frac{x + 1}{x + 3}$ so the numerator has no $x$.
Write the top $x + 1$ as $(x + 3) - 2$: $\frac{(x + 3) - 2}{x + 3}$.
Splitting the fraction gives $1 - \frac{2}{x + 3}$.

Rewrite $\frac{2x}{x - 1}$ as a whole number plus a fraction.
Write the top $2x$ as $2(x - 1) + 2$: $\frac{2(x - 1) + 2}{x - 1}$.
Splitting the fraction gives $2 + \frac{2}{x - 1}$.