Factor polynomial $$ 6x^{2}+51x+63 $$

$$ 6x^{2}+51x+63 = 3( 2x+3 )( x+7 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 6x^{2}+51x+63 = 3 ( 2x^{2}+17x+21 ) $$

Step 2 :

Step 2: 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:

$$ a = 2 , b = 17 ~ \text{ and } ~ c = 21 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 21} $.

$$ a \cdot c = 42 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 42 $ and add to $ b = 17 $.

Step 5: 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 6: Find out which factor pair sums up to $\color{blue}{ b = 17 }$

PRODUCT = 42 and SUM = 17
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 7: Replace middle term $ 17 x $ with $ 14x+3x $:

$$ 2x^{2}+17x+21 = 2x^{2}+14x+3x+21 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 3 $ out of the last two terms.

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

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