Taking common factors

To factor an expression is to rewrite it as things multiplied together.
It undoes expanding: expanding turns $3(2x + 3)$ into $6x + 9$, and factoring turns $6x + 9$ back into $3(2x + 3)$.
Always look first for a greatest common factor — the largest number and variable power shared by every term.

In $6x + 9$, both terms are divisible by $3$, so factor it out: $6x + 9 = 3(2x + 3)$.

The same works with variables: in $2x^3 + 3x^2$, both terms share $x^2$, so $2x^3 + 3x^2 = x^2(2x + 3)$.

Many quadratics need factoring by grouping.
Take $2x^2 + 7x + 5$.
Split the middle term $7x$ into $2x + 5x$, giving $2x^2 + 2x + 5x + 5$.
Group into pairs and factor each: $2x(x + 1) + 5(x + 1)$.
Both pairs share $(x + 1)$, so pull it out: $(x + 1)(2x + 5)$.

To know how to split the middle term, use $a$ and $c$.
Find two numbers that multiply to $a \times c$ and add to $b$.
For $2x^2 + 7x + 5$, $a \times c = 10$ and $b = 7$: the numbers $2$ and $5$ work, which is why we split $7x$ into $2x + 5x$.

Factoring lets you simplify fractions and solve equations.
For example, $\frac{2x + 10}{x^2 - 25}$ factors to $\frac{2(x + 5)}{(x + 5)(x - 5)}$, and the $(x + 5)$ cancels to leave $\frac{2}{x - 5}$.

Factor $8x^2 + 12x$ by pulling out the greatest common factor.
Both terms share $4x$, since $8x^2 = 4x \cdot 2x$ and $12x = 4x \cdot 3$.
So $8x^2 + 12x = 4x(2x + 3)$.

Factor $2x^2 + 7x + 3$ by grouping.
Here $a \times c = 6$ and $b = 7$, so the numbers $6$ and $1$ split the middle term.
$2x^2 + 6x + x + 3$ groups as $2x(x + 3) + 1(x + 3)$.
Both share $(x + 3)$, so it factors to $(x + 3)(2x + 1)$.

Find the greatest common factor of $12x^3y^2$ and $18x^2y^3$.
For the numbers, the GCF of $12$ and $18$ is $6$.
For each variable, take the lower power: $x^2$ and $y^2$.
So the GCF is $6x^2y^2$.