Factor polynomial $$ 5x^{2}+29x+20 $$

$$ 5x^{2}+29x+20 = ( 5x+4 )( x+5 ) $$

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:

$$ a = 5 , b = 29 ~ \text{ and } ~ c = 20 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 20} $.

$$ a \cdot c = 100 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 100 $ and add to $ b = 29 $.

Step 4: All pairs of numbers with a product of $ 100 $ are:

PRODUCT = 100
1     100	-1     -100
2     50	-2     -50
4     25	-4     -25
5     20	-5     -20
10     10	-10     -10

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 29 }$

PRODUCT = 100 and SUM = 29
1     100	-1     -100
2     50	-2     -50
4     25	-4     -25
5     20	-5     -20
10     10	-10     -10

Step 6: Replace middle term $ 29 x $ with $ 25x+4x $:

$$ 5x^{2}+29x+20 = 5x^{2}+25x+4x+20 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 4 $ out of the last two terms.

$$ 5x^{2}+25x+4x+20 = 5x\left(x+5\right) + 4\left(x+5\right) = \left(5x+4\right) \left(x+5\right) $$

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