Factor polynomial $$ 10p^{2}-p-2 $$

$$ 10p^{2}-p-2 = ( 2p-1 )( 5p+2 ) $$

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:

$$ a = 10 , b = -1 ~ \text{ and } ~ c = -2 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = -2} $.

$$ a \cdot c = -20 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -20 $ and add to $ b = -1 $.

Step 4: All pairs of numbers with a product of $ -20 $ are:

PRODUCT = -20
-1     20	1     -20
-2     10	2     -10
-4     5	4     -5

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

PRODUCT = -20 and SUM = -1
-1     20	1     -20
-2     10	2     -10
-4     5	4     -5

Step 6: Replace middle term $ -1 x $ with $ 4x-5x $:

$$ 10x^{2}-x-2 = 10x^{2}+4x-5x-2 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 10x^{2}+4x-5x-2 = 2x\left(5x+2\right) -1\left(5x+2\right) = \left(2x-1\right) \left(5x+2\right) $$

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