Factor polynomial $$ 8x^{2}+14x+5 $$

$$ 8x^{2}+14x+5 = ( 2x+1 )( 4x+5 ) $$

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:

$$ a = 8 , b = 14 ~ \text{ and } ~ c = 5 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = 5} $.

$$ a \cdot c = 40 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 40 $ and add to $ b = 14 $.

Step 4: All pairs of numbers with a product of $ 40 $ are:

PRODUCT = 40
1     40	-1     -40
2     20	-2     -20
4     10	-4     -10
5     8	-5     -8

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 14 }$

PRODUCT = 40 and SUM = 14
1     40	-1     -40
2     20	-2     -20
4     10	-4     -10
5     8	-5     -8

Step 6: Replace middle term $ 14 x $ with $ 10x+4x $:

$$ 8x^{2}+14x+5 = 8x^{2}+10x+4x+5 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 8x^{2}+10x+4x+5 = 2x\left(4x+5\right) + 1\left(4x+5\right) = \left(2x+1\right) \left(4x+5\right) $$

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