Factor polynomial $$ 56x^{2}+39x+4 $$

$$ 56x^{2}+39x+4 = ( 8x+1 )( 7x+4 ) $$

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:

$$ a = 56 , b = 39 ~ \text{ and } ~ c = 4 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 56 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 224 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 224 $ and add to $ b = 39 $.

Step 4: All pairs of numbers with a product of $ 224 $ are:

PRODUCT = 224
1     224	-1     -224
2     112	-2     -112
4     56	-4     -56
7     32	-7     -32
8     28	-8     -28
14     16	-14     -16

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

PRODUCT = 224 and SUM = 39
1     224	-1     -224
2     112	-2     -112
4     56	-4     -56
7     32	-7     -32
8     28	-8     -28
14     16	-14     -16

Step 6: Replace middle term $ 39 x $ with $ 32x+7x $:

$$ 56x^{2}+39x+4 = 56x^{2}+32x+7x+4 $$

Step 7: Apply factoring by grouping. Factor $ 8x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 56x^{2}+32x+7x+4 = 8x\left(7x+4\right) + 1\left(7x+4\right) = \left(8x+1\right) \left(7x+4\right) $$

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