Factor polynomial $$ 6x^{2}+35x+49 $$

$$ 6x^{2}+35x+49 = ( 3x+7 )( 2x+7 ) $$

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:

$$ a = 6 , b = 35 ~ \text{ and } ~ c = 49 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 49} $.

$$ a \cdot c = 294 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 294 $ and add to $ b = 35 $.

Step 4: All pairs of numbers with a product of $ 294 $ are:

PRODUCT = 294
1     294	-1     -294
2     147	-2     -147
3     98	-3     -98
6     49	-6     -49
7     42	-7     -42
14     21	-14     -21

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 35 }$

PRODUCT = 294 and SUM = 35
1     294	-1     -294
2     147	-2     -147
3     98	-3     -98
6     49	-6     -49
7     42	-7     -42
14     21	-14     -21

Step 6: Replace middle term $ 35 x $ with $ 21x+14x $:

$$ 6x^{2}+35x+49 = 6x^{2}+21x+14x+49 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ 7 $ out of the last two terms.

$$ 6x^{2}+21x+14x+49 = 3x\left(2x+7\right) + 7\left(2x+7\right) = \left(3x+7\right) \left(2x+7\right) $$

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