Analogous to integer long division, this method works for any two polynomials. Ensure both are in standard form with zero placeholders for missing terms. The process is a cycle: Divide, Multiply, Subtract, Bring Down.
x + 7
_______
x-2 | x² + 5x + 3
-(x² - 2x)
----------
7x + 3
-(7x - 14)
----------
17
The result is $x+7$ with a remainder of $17$, written as $x+7+\frac{17}{x-2}$.
This is a fast shortcut for dividing by a linear binomial of the form $(x-a)$. It uses only coefficients in a "bring down, multiply and add" pattern. This repetitive process is a great opportunity to practice skills like those honed by spaced repetition.
Example: Divide $(x^2+5x-3)$ by $(x+2)$. Use $a=-2$.
-2 | 1 5 -3
| -2 -6
-------------
1 3 -9
The quotient is $x+3$ and the remainder is $-9$, written as $x+3 - \frac{9}{x+2}$.