Factor polynomial $$ 10x^{2}+9x+2 $$

$$ 10x^{2}+9x+2 = ( 5x+2 )( 2x+1 ) $$

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:

$$ a = 10 , b = 9 ~ \text{ and } ~ c = 2 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = 2} $.

$$ a \cdot c = 20 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 20 $ and add to $ b = 9 $.

Step 4: All pairs of numbers with a product of $ 20 $ are:

PRODUCT = 20
1     20	-1     -20
2     10	-2     -10
4     5	-4     -5

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 9 }$

PRODUCT = 20 and SUM = 9
1     20	-1     -20
2     10	-2     -10
4     5	-4     -5

Step 6: Replace middle term $ 9 x $ with $ 5x+4x $:

$$ 10x^{2}+9x+2 = 10x^{2}+5x+4x+2 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 2 $ out of the last two terms.

$$ 10x^{2}+5x+4x+2 = 5x\left(2x+1\right) + 2\left(2x+1\right) = \left(5x+2\right) \left(2x+1\right) $$

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