Factor polynomial $$ 3x^{2}+22x+7 $$

$$ 3x^{2}+22x+7 = ( 3x+1 )( x+7 ) $$

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:

$$ a = 3 , b = 22 ~ \text{ and } ~ c = 7 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 7} $.

$$ a \cdot c = 21 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 21 $ and add to $ b = 22 $.

Step 4: All pairs of numbers with a product of $ 21 $ are:

PRODUCT = 21
1     21	-1     -21
3     7	-3     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 22 }$

PRODUCT = 21 and SUM = 22
1     21	-1     -21
3     7	-3     -7

Step 6: Replace middle term $ 22 x $ with $ 21x+x $:

$$ 3x^{2}+22x+7 = 3x^{2}+21x+x+7 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 3x^{2}+21x+x+7 = 3x\left(x+7\right) + 1\left(x+7\right) = \left(3x+1\right) \left(x+7\right) $$

Polynomial Factoring Calculator