Factor polynomial $$ 3x^{2}+28x+49 $$

$$ 3x^{2}+28x+49 = ( 3x+7 )( x+7 ) $$

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:

$$ a = 3 , b = 28 ~ \text{ and } ~ c = 49 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 49} $.

$$ a \cdot c = 147 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 147 $ and add to $ b = 28 $.

Step 4: All pairs of numbers with a product of $ 147 $ are:

PRODUCT = 147
1     147	-1     -147
3     49	-3     -49
7     21	-7     -21

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 28 }$

PRODUCT = 147 and SUM = 28
1     147	-1     -147
3     49	-3     -49
7     21	-7     -21

Step 6: Replace middle term $ 28 x $ with $ 21x+7x $:

$$ 3x^{2}+28x+49 = 3x^{2}+21x+7x+49 $$

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

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

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