Step 1 :

After factoring out $ 2 $ we have:

$$ 4x^{2}+22x+10 = 2 ( 2x^{2}+11x+5 ) $$

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:

$$ a = 2 , b = 11 ~ \text{ and } ~ c = 5 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 5} $.

$$ a \cdot c = 10 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 10 $ and add to $ b = 11 $.

Step 5: All pairs of numbers with a product of $ 10 $ are:

PRODUCT = 10
1     10	-1     -10
2     5	-2     -5

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 11 }$

PRODUCT = 10 and SUM = 11
1     10	-1     -10
2     5	-2     -5

Step 7: Replace middle term $ 11 x $ with $ 10x+x $:

$$ 2x^{2}+11x+5 = 2x^{2}+10x+x+5 $$

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

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

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