Factor polynomial $$ 6a^{2}+5a+1 $$

$$ 6a^{2}+5a+1 = ( 3a+1 )( 2a+1 ) $$

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:

$$ a = 6 , b = 5 ~ \text{ and } ~ c = 1 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 6 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 6 $ and add to $ b = 5 $.

Step 4: All pairs of numbers with a product of $ 6 $ are:

PRODUCT = 6
1     6	-1     -6
2     3	-2     -3

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

PRODUCT = 6 and SUM = 5
1     6	-1     -6
2     3	-2     -3

Step 6: Replace middle term $ 5 x $ with $ 3x+2x $:

$$ 6x^{2}+5x+1 = 6x^{2}+3x+2x+1 $$

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

$$ 6x^{2}+3x+2x+1 = 3x\left(2x+1\right) + 1\left(2x+1\right) = \left(3x+1\right) \left(2x+1\right) $$

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