Factor polynomial $$ 10x^{2}-54x-112 $$

$$ 10x^{2}-54x-112 = 2( x-7 )( 5x+8 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 10x^{2}-54x-112 = 2 ( 5x^{2}-27x-56 ) $$

Step 2 :

Step 2: 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:

$$ a = 5 , b = -27 ~ \text{ and } ~ c = -56 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -56} $.

$$ a \cdot c = -280 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -280 $ and add to $ b = -27 $.

Step 5: All pairs of numbers with a product of $ -280 $ are:

PRODUCT = -280
-1     280	1     -280
-2     140	2     -140
-4     70	4     -70
-5     56	5     -56
-7     40	7     -40
-8     35	8     -35
-10     28	10     -28
-14     20	14     -20

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -27 }$

PRODUCT = -280 and SUM = -27
-1     280	1     -280
-2     140	2     -140
-4     70	4     -70
-5     56	5     -56
-7     40	7     -40
-8     35	8     -35
-10     28	10     -28
-14     20	14     -20

Step 7: Replace middle term $ -27 x $ with $ 8x-35x $:

$$ 5x^{2}-27x-56 = 5x^{2}+8x-35x-56 $$

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

$$ 5x^{2}+8x-35x-56 = x\left(5x+8\right) -7\left(5x+8\right) = \left(x-7\right) \left(5x+8\right) $$

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