Factor polynomial $$ 2n^{3}+3n^{2}+n $$

$$ 2n^{3}+3n^{2}+n = n( 2n+1 )( n+1 ) $$

Step 1 :

After factoring out $ n $ we have:

$$ 2n^{3}+3n^{2}+n = n ( 2n^{2}+3n+1 ) $$

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:

$$ a = 2 , b = 3 ~ \text{ and } ~ c = 1 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 2 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 2 $ and add to $ b = 3 $.

Step 5: All pairs of numbers with a product of $ 2 $ are:

PRODUCT = 2
1     2	-1     -2

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 3 }$

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

Step 7: Replace middle term $ 3 x $ with $ 2x+x $:

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

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 1 $ out of the last two terms.

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

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