Step 1 :

After factoring out $ 2 $ we have:

$$ 10x^{2}+24x+8 = 2 ( 5x^{2}+12x+4 ) $$

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:

$$ a = 5 , b = 12 ~ \text{ and } ~ c = 4 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 20 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 20 $ and add to $ b = 12 $.

Step 5: 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 6: Find out which factor pair sums up to $\color{blue}{ b = 12 }$

PRODUCT = 20 and SUM = 12
1     20	-1     -20
2     10	-2     -10
4     5	-4     -5

Step 7: Replace middle term $ 12 x $ with $ 10x+2x $:

$$ 5x^{2}+12x+4 = 5x^{2}+10x+2x+4 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ 2 $ out of the last two terms.

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

Polynomial Factoring Calculator