Factor polynomial $$ 56v^{2}-113v+56 $$

$$ 56v^{2}-113v+56 = ( 7v-8 )( 8v-7 ) $$

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:

$$ a = 56 , b = -113 ~ \text{ and } ~ c = 56 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 56 }$ by the constant term $\color{blue}{c = 56} $.

$$ a \cdot c = 3136 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 3136 $ and add to $ b = -113 $.

Step 4: All pairs of numbers with a product of $ 3136 $ are:

PRODUCT = 3136
1     3136	-1     -3136
2     1568	-2     -1568
4     784	-4     -784
7     448	-7     -448
8     392	-8     -392
14     224	-14     -224
16     196	-16     -196
28     112	-28     -112
32     98	-32     -98
49     64	-49     -64
56     56	-56     -56

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -113 }$

PRODUCT = 3136 and SUM = -113
1     3136	-1     -3136
2     1568	-2     -1568
4     784	-4     -784
7     448	-7     -448
8     392	-8     -392
14     224	-14     -224
16     196	-16     -196
28     112	-28     -112
32     98	-32     -98
49     64	-49     -64
56     56	-56     -56

Step 6: Replace middle term $ -113 x $ with $ -49x-64x $:

$$ 56x^{2}-113x+56 = 56x^{2}-49x-64x+56 $$

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

$$ 56x^{2}-49x-64x+56 = 7x\left(8x-7\right) -8\left(8x-7\right) = \left(7x-8\right) \left(8x-7\right) $$

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