Factor polynomial $$ 2x^{2}-x-1 $$

$$ 2x^{2}-x-1 = ( x-1 )( 2x+1 ) $$

Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 2 , b = -1 ~ \text{ and } ~ c = -1 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -1} $.

$$ a \cdot c = -2 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -2 $ and add to $ b = -1 $.

Step 4: All pairs of numbers with a product of $ -2 $ are:

PRODUCT = -2
-1     2	1     -2

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

PRODUCT = -2 and SUM = -1
-1     2	1     -2

Step 6: Replace middle term $ -1 x $ with $ x-2x $:

$$ 2x^{2}-x-1 = 2x^{2}+x-2x-1 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 2x^{2}+x-2x-1 = x\left(2x+1\right) -1\left(2x+1\right) = \left(x-1\right) \left(2x+1\right) $$

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