Taking out a common factor
Once we have located a common factor in a sum or difference, we can "factor it out" by finding the other factor in each of the summands:
In symbols: Because $a$ is a common factor in $ab \pm ac$, we can take out the common factor $a:$
$\color{#c1026f}{a}\color{#026fc1}{b} \pm \color{#c1026f}{a}\color{#0ea05e}{c} = \color{#c1026f}{a}(\color{#026fc1}{b} \pm \color{#0ea05e}{c}).$
Examples
Because $x$ is a common factor in $2x^2 + x$, we can factor it out:
$2x^2 + x$ \t $\ = \ \color{#c1026f}{(x)}\color{#026fc1}{(2x)} + \color{#c1026f}{(x)}\color{#0ea05e}{(1)}$
\\ \t $\ = \ \color{#c1026f}{x}(\color{#026fc1}{2x} + \color{#0ea05e}{1})$
Because $2y^2$ is a common factor in $6y^4 + 2y^3 - 4y^2$, we can factor it out:
$6y^4 + 2y^3 - 4y^2$ \t $\ = \ \color{#c1026f}{(2y^2)}\color{#026fc1}{(y^2)} + \color{#c1026f}{(2y^2)}\color{#0ea05e}{(y)} - \color{#c1026f}{(2y^2)}\color{#a05eae}{(2)}$
\\ \t $\ = \ \color{#c1026f}{2y^2}(\color{#026fc1}{y^2} + \color{#0ea05e}{y} - \color{#a05eae}{2})$
In the remaining examples we will leave out the middle step (do that mentally!)
Because $x$ is a common factor in $2x^2y+6xy^2-6x^2y^2$, we can factor it out:
$2x^2y+xy^2-x^2y^2 \ = \ x(2xy+6y^2-6xy^2)$
However, $2xy$ is another common factor in $2x^2y+6xy^2-6x^2y^2$; it is the
greatest common factor:
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Its coefficient is positive.
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It cannot be multiplied by anything except $-1$ and still remain a factor.
So, we can also write
$2x^2y+6xy^2-6x^2y^2 \ = \ 2xy(x+3y-3xy). \qquad$ \t