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:

$$ a = 24 , b = 19 ~ \text{ and } ~ c = 2 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 24 }$ by the constant term $\color{blue}{c = 2} $.

$$ a \cdot c = 48 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 48 $ and add to $ b = 19 $.

Step 4: All pairs of numbers with a product of $ 48 $ are:

PRODUCT = 48
1     48	-1     -48
2     24	-2     -24
3     16	-3     -16
4     12	-4     -12
6     8	-6     -8

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 19 }$

PRODUCT = 48 and SUM = 19
1     48	-1     -48
2     24	-2     -24
3     16	-3     -16
4     12	-4     -12
6     8	-6     -8

Step 6: Replace middle term $ 19 x $ with $ 16x+3x $:

$$ 24x^{2}+19x+2 = 24x^{2}+16x+3x+2 $$

Step 7: Apply factoring by grouping. Factor $ 8x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 24x^{2}+16x+3x+2 = 8x\left(3x+2\right) + 1\left(3x+2\right) = \left(8x+1\right) \left(3x+2\right) $$

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