Step 1 :

After factoring out $ 4 $ we have:

$$ 44p^{2}+52p+8 = 4 ( 11p^{2}+13p+2 ) $$

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:

$$ a = 11 , b = 13 ~ \text{ and } ~ c = 2 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 11 }$ by the constant term $\color{blue}{c = 2} $.

$$ a \cdot c = 22 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 22 $ and add to $ b = 13 $.

Step 5: All pairs of numbers with a product of $ 22 $ are:

PRODUCT = 22
1     22	-1     -22
2     11	-2     -11

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 13 }$

PRODUCT = 22 and SUM = 13
1     22	-1     -22
2     11	-2     -11

Step 7: Replace middle term $ 13 x $ with $ 11x+2x $:

$$ 11x^{2}+13x+2 = 11x^{2}+11x+2x+2 $$

Step 8: Apply factoring by grouping. Factor $ 11x $ out of the first two terms and $ 2 $ out of the last two terms.

$$ 11x^{2}+11x+2x+2 = 11x\left(x+1\right) + 2\left(x+1\right) = \left(11x+2\right) \left(x+1\right) $$

Polynomial Factoring Calculator