Factor polynomial $$ 3x^{2}+13x+14 $$

$$ 3x^{2}+13x+14 = ( x+2 )( 3x+7 ) $$

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:

$$ a = 3 , b = 13 ~ \text{ and } ~ c = 14 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 14} $.

$$ a \cdot c = 42 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 42 $ and add to $ b = 13 $.

Step 4: All pairs of numbers with a product of $ 42 $ are:

PRODUCT = 42
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 13 }$

PRODUCT = 42 and SUM = 13
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 6: Replace middle term $ 13 x $ with $ 7x+6x $:

$$ 3x^{2}+13x+14 = 3x^{2}+7x+6x+14 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ 2 $ out of the last two terms.

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

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