Factor polynomial $$ 5z^{2}+22z+8 $$

$$ 5z^{2}+22z+8 = ( 5z+2 )( z+4 ) $$

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:

$$ a = 5 , b = 22 ~ \text{ and } ~ c = 8 $$

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

$$ a \cdot c = 40 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 40 $ and add to $ b = 22 $.

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 $\color{blue}{ b = 22 }$

PRODUCT = 40 and SUM = 22
1     40	-1     -40
2     20	-2     -20
4     10	-4     -10
5     8	-5     -8

Step 6: Replace middle term $ 22 x $ with $ 20x+2x $:

$$ 5x^{2}+22x+8 = 5x^{2}+20x+2x+8 $$

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

$$ 5x^{2}+20x+2x+8 = 5x\left(x+4\right) + 2\left(x+4\right) = \left(5x+2\right) \left(x+4\right) $$

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