Factor polynomial $$ 14v^{2}+92v+48 $$

$$ 14v^{2}+92v+48 = 2( 7v+4 )( v+6 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 14v^{2}+92v+48 = 2 ( 7v^{2}+46v+24 ) $$

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:

$$ a = 7 , b = 46 ~ \text{ and } ~ c = 24 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 24} $.

$$ a \cdot c = 168 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 168 $ and add to $ b = 46 $.

Step 5: All pairs of numbers with a product of $ 168 $ are:

PRODUCT = 168
1     168	-1     -168
2     84	-2     -84
3     56	-3     -56
4     42	-4     -42
6     28	-6     -28
7     24	-7     -24
8     21	-8     -21
12     14	-12     -14

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 46 }$

PRODUCT = 168 and SUM = 46
1     168	-1     -168
2     84	-2     -84
3     56	-3     -56
4     42	-4     -42
6     28	-6     -28
7     24	-7     -24
8     21	-8     -21
12     14	-12     -14

Step 7: Replace middle term $ 46 x $ with $ 42x+4x $:

$$ 7x^{2}+46x+24 = 7x^{2}+42x+4x+24 $$

Step 8: Apply factoring by grouping. Factor $ 7x $ out of the first two terms and $ 4 $ out of the last two terms.

$$ 7x^{2}+42x+4x+24 = 7x\left(x+6\right) + 4\left(x+6\right) = \left(7x+4\right) \left(x+6\right) $$

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