Factor polynomial $$ 40x^{2}-39x-40 $$

$$ 40x^{2}-39x-40 = ( 5x-8 )( 8x+5 ) $$

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:

$$ a = 40 , b = -39 ~ \text{ and } ~ c = -40 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 40 }$ by the constant term $\color{blue}{c = -40} $.

$$ a \cdot c = -1600 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -1600 $ and add to $ b = -39 $.

Step 4: All pairs of numbers with a product of $ -1600 $ are:

PRODUCT = -1600
-1     1600	1     -1600
-2     800	2     -800
-4     400	4     -400
-5     320	5     -320
-8     200	8     -200
-10     160	10     -160
-16     100	16     -100
-20     80	20     -80
-25     64	25     -64
-32     50	32     -50
-40     40	40     -40

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -39 }$

PRODUCT = -1600 and SUM = -39
-1     1600	1     -1600
-2     800	2     -800
-4     400	4     -400
-5     320	5     -320
-8     200	8     -200
-10     160	10     -160
-16     100	16     -100
-20     80	20     -80
-25     64	25     -64
-32     50	32     -50
-40     40	40     -40

Step 6: Replace middle term $ -39 x $ with $ 25x-64x $:

$$ 40x^{2}-39x-40 = 40x^{2}+25x-64x-40 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -8 $ out of the last two terms.

$$ 40x^{2}+25x-64x-40 = 5x\left(8x+5\right) -8\left(8x+5\right) = \left(5x-8\right) \left(8x+5\right) $$

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