Factor polynomial $$ 35x^{2}+47x+6 $$

$$ 35x^{2}+47x+6 = ( 7x+1 )( 5x+6 ) $$

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:

$$ a = 35 , b = 47 ~ \text{ and } ~ c = 6 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 35 }$ by the constant term $\color{blue}{c = 6} $.

$$ a \cdot c = 210 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 210 $ and add to $ b = 47 $.

Step 4: All pairs of numbers with a product of $ 210 $ are:

PRODUCT = 210
1     210	-1     -210
2     105	-2     -105
3     70	-3     -70
5     42	-5     -42
6     35	-6     -35
7     30	-7     -30
10     21	-10     -21
14     15	-14     -15

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 47 }$

PRODUCT = 210 and SUM = 47
1     210	-1     -210
2     105	-2     -105
3     70	-3     -70
5     42	-5     -42
6     35	-6     -35
7     30	-7     -30
10     21	-10     -21
14     15	-14     -15

Step 6: Replace middle term $ 47 x $ with $ 42x+5x $:

$$ 35x^{2}+47x+6 = 35x^{2}+42x+5x+6 $$

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

$$ 35x^{2}+42x+5x+6 = 7x\left(5x+6\right) + 1\left(5x+6\right) = \left(7x+1\right) \left(5x+6\right) $$

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